area y=x+x^2+x^3,x=1

{ "query": { "display": "area $$y=x+x^{2}+x^{3},\\:x=1$$", "symbolab_question": "INTEGRAL_APPLICATION#area y=x+x^{2}+x^{3},x=1" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integral Applications", "subTopic": "Area under the curve", "default": "\\frac{13}{12}", "decimal": "1.08333…" }, "steps": { "type": "interim", "title": "The area between the curves $$y=x+x^{2}+x^{3}\\:$$and $$x=1:{\\quad}\\frac{13}{12}$$", "steps": [ { "type": "definition", "title": "Area under a curve definition", "text": "The area under a curve is the area between a curve $$f\\left(x\\right)\\:$$and the x-axis on an interval $$[a,\\:b]\\:$$given by<br/>$$A=\\int_{a}^{b}|f\\left(x\\right)|dx$$" }, { "type": "interim", "title": "Apply the area formula:$${\\quad}\\int_{0}^{1}\\left|x+x^{2}+x^{3}\\right|dx$$", "steps": [ { "type": "step", "primary": "$$f\\left(x\\right)=x+x^{2}+x^{3}$$" }, { "type": "interim", "title": "Find intersection points:$${\\quad}x=0$$", "steps": [ { "type": "step", "primary": "To find the intersection points solve $$f\\left(x\\right)=0$$", "result": "x+x^{2}+x^{3}=0" }, { "type": "interim", "title": "$$x+x^{2}+x^{3}=0{\\quad:\\quad}x=0$$", "input": "x+x^{2}+x^{3}=0", "result": "x=0", "steps": [ { "type": "interim", "title": "Factor $$x+x^{2}+x^{3}:{\\quad}x\\left(x^{2}+x+1\\right)$$", "input": "x+x^{2}+x^{3}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$", "$$x^{3}=x^{2}x$$" ], "result": "=x^{2}x+xx+x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x^{2}+x+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Factor Specific 1Eq" } }, { "type": "step", "result": "x\\left(x^{2}+x+1\\right)=0" }, { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$", "result": "x=0\\lor\\:x^{2}+x+1=0" }, { "type": "interim", "title": "Solve $$x^{2}+x+1=0:{\\quad}$$No Solution for $$x\\in\\mathbb{R}$$", "input": "x^{2}+x+1=0", "steps": [ { "type": "interim", "title": "Discriminant $$x^{2}+x+1=0:{\\quad}-3$$", "input": "x^{2}+x+1=0", "steps": [ { "type": "step", "primary": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the discriminant is $$b^2-4ac$$", "secondary": [ "For $${\\quad}a=1,\\:b=1,\\:c=1{:\\quad}1^{2}-4\\cdot\\:1\\cdot\\:1$$" ], "result": "1^{2}-4\\cdot\\:1\\cdot\\:1" }, { "type": "interim", "title": "Expand $$1^{2}-4\\cdot\\:1\\cdot\\:1:{\\quad}-3$$", "input": "1^{2}-4\\cdot\\:1\\cdot\\:1", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=1-4\\cdot\\:1\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1\\cdot\\:1=4$$", "result": "=1-4" }, { "type": "step", "primary": "Subtract the numbers: $$1-4=-3$$", "result": "=-3" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7dNSByaY+kcwSxzDNBdAoWUyltbx1ORuo8uWCi9bAKvjeXCIFfkXoRCDVXhjJDXGVs7PdN8ocStDSJ08Eeyb0kcjP9vZe0h5cDZXk0KEZ9KgScn/M2sLYBGgxClqzCxM2aNh7RL/BTBlIs5N38wwNbAR3EwSCCc/8bX4e0PXgmnA=" } }, { "type": "step", "result": "-3" } ], "meta": { "interimType": "Discriminant Title 1Eq" } }, { "type": "step", "primary": "Discriminant cannot be negative for $$x\\in\\mathbb{R}$$" }, { "type": "step", "primary": "The solution is", "result": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The solution is", "result": "x=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } } ], "meta": { "interimType": "Find Intersection Title 0Eq" } }, { "type": "step", "primary": "Therefore", "secondary": [ "$$a=0,\\:b=1$$" ] }, { "type": "step", "result": "=\\int_{0}^{1}\\left|x+x^{2}+x^{3}\\right|dx" } ], "meta": { "interimType": "Apply Area Formula 0Eq" } }, { "type": "interim", "title": "Solve $$\\int_{0}^{1}\\left|x+x^{2}+x^{3}\\right|dx:{\\quad}\\frac{13}{12}$$", "input": "\\int_{0}^{1}\\left|x+x^{2}+x^{3}\\right|dx", "steps": [ { "type": "interim", "title": "Eliminate Absolutes", "input": "\\int_{0}^{1}\\left|x+x^{2}+x^{3}\\right|dx", "result": "=\\int_{0}^{1}x+x^{2}+x^{3}dx", "steps": [ { "type": "step", "primary": "Find the equivalent expressions to $$\\left|x+x^{2}+x^{3}\\right|$$ at $$0\\le\\:x\\le\\:1$$ without the absolutes" }, { "type": "step", "primary": "$$0\\le\\:x\\le\\:1:{\\quad}x+x^{2}+x^{3}$$" }, { "type": "step", "result": "=\\int_{0}^{1}x+x^{2}+x^{3}dx" } ], "meta": { "interimType": "Eliminate Absolutes Integral 2Eq" } }, { "type": "step", "primary": "Apply the Sum Rule: $$\\int{f\\left(x\\right){\\pm}g\\left(x\\right)}dx=\\int{f\\left(x\\right)}dx{\\pm}\\int{g\\left(x\\right)}dx$$", "result": "=\\int_{0}^{1}xdx+\\int_{0}^{1}x^{2}dx+\\int_{0}^{1}x^{3}dx" }, { "type": "interim", "title": "$$\\int_{0}^{1}xdx=\\frac{1}{2}$$", "input": "\\int_{0}^{1}xdx", "steps": [ { "type": "interim", "title": "Apply the Power Rule", "input": "\\int_{0}^{1}xdx", "result": "=[\\frac{x^{2}}{2}]_{0}^{1}", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=[\\frac{x^{1+1}}{1+1}]_{0}^{1}" }, { "type": "interim", "title": "Simplify $$\\frac{x^{1+1}}{1+1}:{\\quad}\\frac{x^{2}}{2}$$", "input": "\\frac{x^{1+1}}{1+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=\\frac{x^{2}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=[\\frac{x^{2}}{2}]_{0}^{1}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7+Q+W0W2FArTKy+svCrd1QMsjvX7KVUO/AeCFSId4S33HipIftvBYl8MvlbM/MS0IniX35dQ/h01lIvxamZtt5P0OFtj1jyjvPrkcLqztccOf5UEZ2/O3gFmSOyfIvXNqdbA+zX4bD3u3gx65o2NJhN+u9S1UanyCrDStYeLlcNDOFynCe2Jk2u4EAkbH+yVgg==" } }, { "type": "interim", "title": "Compute the boundaries:$${\\quad}\\frac{1}{2}$$", "input": "[\\frac{x^{2}}{2}]_{0}^{1}", "steps": [ { "type": "step", "primary": "$$\\int_{a}^{b}{f\\left(x\\right)dx}=F\\left(b\\right)-F\\left(a\\right)=\\lim_{x\\to\\:b-}\\left(F\\left(x\\right)\\right)-\\lim_{x\\to\\:a+}\\left(F\\left(x\\right)\\right)$$" }, { "type": "interim", "title": "$$\\lim_{x\\to\\:0+}\\left(\\frac{x^{2}}{2}\\right)=0$$", "input": "\\lim_{x\\to\\:0+}\\left(\\frac{x^{2}}{2}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=0$$", "result": "=\\frac{0^{2}}{2}", "meta": { "title": { "extension": "Limit properties - if the limit of f(x), and g(x) exists, then:<br/>$$\\bullet\\quad\\lim_{x\\to\\:a}\\left(x\\right)=a$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[\\left(f\\left(x\\right)\\right)^c]=\\left(\\lim_{x\\to{a}}{f\\left(x\\right)}\\right)^c$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\cdot{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\cdot\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:$$where $$\\lim_{x\\to{a}}g\\left(x\\right)\\neq0$$" } } }, { "type": "interim", "title": "Simplify $$\\frac{0^{2}}{2}:{\\quad}0$$", "input": "\\frac{0^{2}}{2}", "result": "=0", "steps": [ { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "secondary": [ "$$0^{2}=0$$" ], "result": "=\\frac{0}{2}" }, { "type": "step", "primary": "Apply rule $$\\frac{0}{a}=0,\\:a\\ne\\:0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nteUyMJWMp4mm/uXtkFpCVnyYRz18HvB+rp63mPitc8E5aqGN/sLZfeoFZRwtGLqP8vQyhiD4JSfqjIvcQ7timkSOxgqdB0M/sw8Nt2sXXR3DTZYr1PJ9/OYrvIJiwrr+i0Ux3lprvX50CFfl5rrAQ==" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:1-}\\left(\\frac{x^{2}}{2}\\right)=\\frac{1}{2}$$", "input": "\\lim_{x\\to\\:1-}\\left(\\frac{x^{2}}{2}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=1$$", "result": "=\\frac{1^{2}}{2}", "meta": { "title": { "extension": "Limit properties - if the limit of f(x), and g(x) exists, then:<br/>$$\\bullet\\quad\\lim_{x\\to\\:a}\\left(x\\right)=a$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[\\left(f\\left(x\\right)\\right)^c]=\\left(\\lim_{x\\to{a}}{f\\left(x\\right)}\\right)^c$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\cdot{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\cdot\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:$$where $$\\lim_{x\\to{a}}g\\left(x\\right)\\neq0$$" } } }, { "type": "step", "primary": "Simplify", "result": "=\\frac{1}{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=\\frac{1}{2}-0" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{1}{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "interimType": "Integral Definite Limit Boundaries 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7+Q+W0W2FArTKy+svCrd1QOPVo8rHFjKlUhEt/c08PpWH/+9ZCmtxJEW2yM0lAeSa4tu9GwZ3vkMP5rfw6uavIH/P/+v51eTuH2/F4MIu5mqU/td0/rq+faUMiA6Cy3iZF1qkcD3DkFSvldxTLmJepCJqVxX90jlMfh9fKn6dzC4" } }, { "type": "step", "result": "=\\frac{1}{2}" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int_{0}^{1}x^{2}dx=\\frac{1}{3}$$", "input": "\\int_{0}^{1}x^{2}dx", "steps": [ { "type": "interim", "title": "Apply the Power Rule", "input": "\\int_{0}^{1}x^{2}dx", "result": "=[\\frac{x^{3}}{3}]_{0}^{1}", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=[\\frac{x^{2+1}}{2+1}]_{0}^{1}" }, { "type": "interim", "title": "Simplify $$\\frac{x^{2+1}}{2+1}:{\\quad}\\frac{x^{3}}{3}$$", "input": "\\frac{x^{2+1}}{2+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=\\frac{x^{3}}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=[\\frac{x^{3}}{3}]_{0}^{1}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7w92TMsWqGt31S9pHaB+pEqTdaV09PMxEKZ9FieghTFwnz9JShbQBLExa7JLGJEbAosjLe8tD9HbrkG8vq6q9jgNBQ+F+TDBmu5NGuvi/fcbiBEiKnSx326UoVFHldUEqEUqTd96MWTKI6Kr2Ib0iQBbgTZHtHmQDZkDHT6lbFNTOZoJMX7Twq11mSDHSln+YA==" } }, { "type": "interim", "title": "Compute the boundaries:$${\\quad}\\frac{1}{3}$$", "input": "[\\frac{x^{3}}{3}]_{0}^{1}", "steps": [ { "type": "step", "primary": "$$\\int_{a}^{b}{f\\left(x\\right)dx}=F\\left(b\\right)-F\\left(a\\right)=\\lim_{x\\to\\:b-}\\left(F\\left(x\\right)\\right)-\\lim_{x\\to\\:a+}\\left(F\\left(x\\right)\\right)$$" }, { "type": "interim", "title": "$$\\lim_{x\\to\\:0+}\\left(\\frac{x^{3}}{3}\\right)=0$$", "input": "\\lim_{x\\to\\:0+}\\left(\\frac{x^{3}}{3}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=0$$", "result": "=\\frac{0^{3}}{3}", "meta": { "title": { "extension": "Limit properties - if the limit of f(x), and g(x) exists, then:<br/>$$\\bullet\\quad\\lim_{x\\to\\:a}\\left(x\\right)=a$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[\\left(f\\left(x\\right)\\right)^c]=\\left(\\lim_{x\\to{a}}{f\\left(x\\right)}\\right)^c$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\cdot{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\cdot\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:$$where $$\\lim_{x\\to{a}}g\\left(x\\right)\\neq0$$" } } }, { "type": "interim", "title": "Simplify $$\\frac{0^{3}}{3}:{\\quad}0$$", "input": "\\frac{0^{3}}{3}", "result": "=0", "steps": [ { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "secondary": [ "$$0^{3}=0$$" ], "result": "=\\frac{0}{3}" }, { "type": "step", "primary": "Apply rule $$\\frac{0}{a}=0,\\:a\\ne\\:0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qBRpSLsqhPH9RPF5elXY++T0JTwwJam8DFLG9sjVKUME5aqGN/sLZfeoFZRwtGLqP8vQyhiD4JSfqjIvcQ7timkSOxgqdB0M/sw8Nt2sXXRjRfJ/eTycHczxN2mOTuFsEk7YwRD0hobDQfAqSn6vxw==" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:1-}\\left(\\frac{x^{3}}{3}\\right)=\\frac{1}{3}$$", "input": "\\lim_{x\\to\\:1-}\\left(\\frac{x^{3}}{3}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=1$$", "result": "=\\frac{1^{3}}{3}", "meta": { "title": { "extension": "Limit properties - if the limit of f(x), and g(x) exists, then:<br/>$$\\bullet\\quad\\lim_{x\\to\\:a}\\left(x\\right)=a$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[\\left(f\\left(x\\right)\\right)^c]=\\left(\\lim_{x\\to{a}}{f\\left(x\\right)}\\right)^c$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\cdot{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\cdot\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:$$where $$\\lim_{x\\to{a}}g\\left(x\\right)\\neq0$$" } } }, { "type": "step", "primary": "Simplify", "result": "=\\frac{1}{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=\\frac{1}{3}-0" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{1}{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "interimType": "Integral Definite Limit Boundaries 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7w92TMsWqGt31S9pHaB+pEom9V/Bk1ibjaeyWJhgrw41uQ4kmq/soqHzBZwEaFJcUqd+Ty23jxMZLX6RHrRLaRGgtm0MPhUSpjQTUqAbvqvJi4n0v56P5SoFAesqLAyUYwanIPMU//UDMHbnrRFJUfzqvuVTzkV43woA0R3vH6y9" } }, { "type": "step", "result": "=\\frac{1}{3}" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int_{0}^{1}x^{3}dx=\\frac{1}{4}$$", "input": "\\int_{0}^{1}x^{3}dx", "steps": [ { "type": "interim", "title": "Apply the Power Rule", "input": "\\int_{0}^{1}x^{3}dx", "result": "=[\\frac{x^{4}}{4}]_{0}^{1}", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=[\\frac{x^{3+1}}{3+1}]_{0}^{1}" }, { "type": "interim", "title": "Simplify $$\\frac{x^{3+1}}{3+1}:{\\quad}\\frac{x^{4}}{4}$$", "input": "\\frac{x^{3+1}}{3+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=\\frac{x^{4}}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=[\\frac{x^{4}}{4}]_{0}^{1}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s714ZdI58A6px+C7dE/wSh3mTdaV09PMxEKZ9FieghTFwnz9JShbQBLExa7JLGJEbAosjLe8tD9HbrkG8vq6q9jjtfmb9XXfd5ttvMBzR/l4YZxJnQY1+wRgPIKU6QDpxvEUqTd96MWTKI6Kr2Ib0iQBbgTZHtHmQDZkDHT6lbFNTOZoJMX7Twq11mSDHSln+YA==" } }, { "type": "interim", "title": "Compute the boundaries:$${\\quad}\\frac{1}{4}$$", "input": "[\\frac{x^{4}}{4}]_{0}^{1}", "steps": [ { "type": "step", "primary": "$$\\int_{a}^{b}{f\\left(x\\right)dx}=F\\left(b\\right)-F\\left(a\\right)=\\lim_{x\\to\\:b-}\\left(F\\left(x\\right)\\right)-\\lim_{x\\to\\:a+}\\left(F\\left(x\\right)\\right)$$" }, { "type": "interim", "title": "$$\\lim_{x\\to\\:0+}\\left(\\frac{x^{4}}{4}\\right)=0$$", "input": "\\lim_{x\\to\\:0+}\\left(\\frac{x^{4}}{4}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=0$$", "result": "=\\frac{0^{4}}{4}", "meta": { "title": { "extension": "Limit properties - if the limit of f(x), and g(x) exists, then:<br/>$$\\bullet\\quad\\lim_{x\\to\\:a}\\left(x\\right)=a$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[\\left(f\\left(x\\right)\\right)^c]=\\left(\\lim_{x\\to{a}}{f\\left(x\\right)}\\right)^c$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\cdot{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\cdot\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:$$where $$\\lim_{x\\to{a}}g\\left(x\\right)\\neq0$$" } } }, { "type": "interim", "title": "Simplify $$\\frac{0^{4}}{4}:{\\quad}0$$", "input": "\\frac{0^{4}}{4}", "result": "=0", "steps": [ { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "secondary": [ "$$0^{4}=0$$" ], "result": "=\\frac{0}{4}" }, { "type": "step", "primary": "Apply rule $$\\frac{0}{a}=0,\\:a\\ne\\:0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7SIJzsSdJI+WyHot1aBR+lGFExV4mFNoq+OGxoJmhOAkE5aqGN/sLZfeoFZRwtGLqP8vQyhiD4JSfqjIvcQ7timkSOxgqdB0M/sw8Nt2sXXRGCE6eBJR17v41x11J2XrZzpWlbt/AlNC3EwOtC8OHxw==" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:1-}\\left(\\frac{x^{4}}{4}\\right)=\\frac{1}{4}$$", "input": "\\lim_{x\\to\\:1-}\\left(\\frac{x^{4}}{4}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=1$$", "result": "=\\frac{1^{4}}{4}", "meta": { "title": { "extension": "Limit properties - if the limit of f(x), and g(x) exists, then:<br/>$$\\bullet\\quad\\lim_{x\\to\\:a}\\left(x\\right)=a$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[\\left(f\\left(x\\right)\\right)^c]=\\left(\\lim_{x\\to{a}}{f\\left(x\\right)}\\right)^c$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\cdot{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\cdot\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:$$where $$\\lim_{x\\to{a}}g\\left(x\\right)\\neq0$$" } } }, { "type": "step", "primary": "Simplify", "result": "=\\frac{1}{4}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=\\frac{1}{4}-0" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{1}{4}", "meta": { "solvingClass": "Solver" } } ], "meta": { "interimType": "Integral Definite Limit Boundaries 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s714ZdI58A6px+C7dE/wSh3km9V/Bk1ibjaeyWJhgrw41uQ4kmq/soqHzBZwEaFJcUhrDTchWYTtEunqHYbO9i/2gtm0MPhUSpjQTUqAbvqvJi4n0v56P5SoFAesqLAyUYwanIPMU//UDMHbnrRFJUfzqvuVTzkV43woA0R3vH6y9" } }, { "type": "step", "result": "=\\frac{1}{4}" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{4}" }, { "type": "interim", "title": "Simplify $$\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{4}:{\\quad}\\frac{13}{12}$$", "input": "\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{4}", "result": "=\\frac{13}{12}", "steps": [ { "type": "interim", "title": "Least Common Multiplier of $$2,\\:3,\\:4:{\\quad}12$$", "input": "2,\\:3,\\:4", "steps": [ { "type": "definition", "title": "Least Common Multiplier (LCM)", "text": "The LCM of $$a,\\:b$$ is the smallest positive number that is divisible by both $$a$$ and $$b$$" }, { "type": "interim", "title": "Prime factorization of $$2:{\\quad}2$$", "input": "2", "steps": [ { "type": "step", "primary": "$$2$$ is a prime number, therefore no factorization is possible", "result": "=2" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRl8ZboA8wPLg0yhI4RzfjFw/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp1+G9v2aKasChgV65VW8cTW" } }, { "type": "interim", "title": "Prime factorization of $$3:{\\quad}3$$", "input": "3", "steps": [ { "type": "step", "primary": "$$3$$ is a prime number, therefore no factorization is possible", "result": "=3" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRlqnfsqoQ6VBiS8EyG3E6Oc/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp39nbJbLYrlgLb4BA6ndvX8" } }, { "type": "interim", "title": "Prime factorization of $$4:{\\quad}2\\cdot\\:2$$", "input": "4", "steps": [ { "type": "step", "primary": "$$4\\:$$divides by $$2\\quad\\:4=2\\cdot\\:2$$", "result": "=2\\cdot\\:2" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRsG/uC0ndYtZpJL4uAxK7FI/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp39fF/zAtU5baHQ1hwgXA+n" } }, { "type": "step", "primary": "Compute a number comprised of factors that appear in at least one of the following:<br/>$$2,\\:3,\\:4$$", "result": "=2\\cdot\\:2\\cdot\\:3" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2\\cdot\\:3=12$$", "result": "=12" } ], "meta": { "solvingClass": "LCM", "interimType": "LCM Top 1Eq" } }, { "type": "interim", "title": "Adjust Fractions based on the LCM", "steps": [ { "type": "step", "primary": "Multiply each numerator by the same amount needed to multiply its<br/>corresponding denominator to turn it into the LCM $$12$$" }, { "type": "step", "primary": "For $$\\frac{1}{2}:\\:$$multiply the denominator and numerator by $$6$$", "result": "\\frac{1}{2}=\\frac{1\\cdot\\:6}{2\\cdot\\:6}=\\frac{6}{12}" }, { "type": "step", "primary": "For $$\\frac{1}{3}:\\:$$multiply the denominator and numerator by $$4$$", "result": "\\frac{1}{3}=\\frac{1\\cdot\\:4}{3\\cdot\\:4}=\\frac{4}{12}" }, { "type": "step", "primary": "For $$\\frac{1}{4}:\\:$$multiply the denominator and numerator by $$3$$", "result": "\\frac{1}{4}=\\frac{1\\cdot\\:3}{4\\cdot\\:3}=\\frac{3}{12}" } ], "meta": { "interimType": "LCD Adjust Fractions 1Eq" } }, { "type": "step", "result": "=\\frac{6}{12}+\\frac{4}{12}+\\frac{3}{12}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{6+4+3}{12}" }, { "type": "step", "primary": "Add the numbers: $$6+4+3=13$$", "result": "=\\frac{13}{12}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/IqmB+KRU+T1KryXQuitcXH25cOZuRLvqYaQC6Q64SVFUyHKlhZauCyfqfKWDVNsdYPfXQvX4/bINBB8wSEQ0XGSIsxobm1ZpQLMBOwHqWwWkHXm89fGaKpG7Uu204cd7kAjP76qW66lOUsURwT0nb4sd9vpffgNCYuOxFpaSofc8u0JRPjzbWeW3iNb64MR1zmgcD+ls8u2Jqrz0f4YEg==" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The area is:", "result": "=\\frac{13}{12}" } ] }, "plot_output": { "meta": { "plotInfo": { "variable": "y", "plotRequest": "yes" }, "showViewLarger": true } } }