area 2x,x=3

{ "query": { "display": "area $$2x,\\:x=3$$", "symbolab_question": "INTEGRAL_APPLICATION#area 2x,x=3" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integral Applications", "subTopic": "Area under the curve", "default": "9" }, "steps": { "type": "interim", "title": "The area between the curves $$f\\left(x\\right)=2x\\:$$and $$x=3:{\\quad}9$$", "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:$${\\quad}\\int_{0}^{3}2\\left|x\\right|dx$$", "steps": [ { "type": "step", "primary": "$$f\\left(x\\right)=2x$$" }, { "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": "2x=0" }, { "type": "interim", "title": "$$2x=0{\\quad:\\quad}x=0$$", "input": "2x=0", "result": "x=0", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=0", "result": "x=0", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{0}{2}" }, { "type": "step", "primary": "Simplify", "result": "x=0" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } } ], "meta": { "interimType": "Find Intersection Title 0Eq" } }, { "type": "step", "primary": "Therefore", "secondary": [ "$$a=0,\\:b=3$$" ] }, { "type": "step", "result": "=\\int_{0}^{3}\\left|2x\\right|dx" }, { "type": "step", "primary": "Simplify" }, { "type": "step", "result": "=\\int_{0}^{3}2\\left|x\\right|dx" } ], "meta": { "interimType": "Apply Area Formula 0Eq" } }, { "type": "interim", "title": "Solve $$\\int_{0}^{3}2\\left|x\\right|dx:{\\quad}9$$", "input": "\\int_{0}^{3}2\\left|x\\right|dx", "steps": [ { "type": "interim", "title": "Eliminate Absolutes", "input": "\\int_{0}^{3}2\\left|x\\right|dx", "result": "=\\int_{0}^{3}2xdx", "steps": [ { "type": "step", "primary": "Find the equivalent expressions to $$2\\left|x\\right|$$ at $$0\\le\\:x\\le\\:3$$ without the absolutes" }, { "type": "step", "primary": "$$0\\le\\:x\\le\\:3:{\\quad}2x$$" }, { "type": "step", "result": "=\\int_{0}^{3}2xdx" } ], "meta": { "interimType": "Eliminate Absolutes Integral 2Eq" } }, { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=2\\cdot\\:\\int_{0}^{3}xdx" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int_{0}^{3}xdx", "result": "=2[\\frac{x^{2}}{2}]_{0}^{3}", "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}^{3}" }, { "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}^{3}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s73eIza12x517zqyy2AdD8WwsjvX7KVUO/AeCFSId4S33HipIftvBYl8MvlbM/MS0IniX35dQ/h01lIvxamZtt5P0OFtj1jyjvPrkcLqztccOZ4EqGUH6luvzhLyP4MrprdbA+zX4bD3u3gx65o2NJhN+u9S1UanyCrDStYeLlcNDOFynCe2Jk2u4EAkbH+yVgg==" } }, { "type": "interim", "title": "Compute the boundaries:$${\\quad}\\frac{9}{2}$$", "input": "[\\frac{x^{2}}{2}]_{0}^{3}", "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": "$$\\lim_{x\\to\\:3-}\\left(\\frac{x^{2}}{2}\\right)=\\frac{9}{2}$$", "input": "\\lim_{x\\to\\:3-}\\left(\\frac{x^{2}}{2}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=3$$", "result": "=\\frac{3^{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{9}{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=\\frac{9}{2}-0" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{9}{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "interimType": "Integral Definite Limit Boundaries 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7wSe6QRBX4py1NvBE779HsmMzLE9R3ZOJUe+Rh6pK1ptj1aPKxxYypVIRLf3NPD6Vh//vWQprcSRFtsjNJQHkmsDqM/FZp+k6cBZGeSeUXb0/z//r+dXk7h9vxeDCLuZqlP7XdP66vn2lDIgOgst4mRdapHA9w5BUr5XcUy5iXqQialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "step", "result": "=2\\cdot\\:\\frac{9}{2}" }, { "type": "step", "primary": "Simplify", "result": "=9" } ], "meta": { "solvingClass": "Integrals", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The area is:", "result": "=9" } ] }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "yes" }, "showViewLarger": true } } }