area y=4x^2,y=x^2+6

{ "query": { "display": "area $$y=4x^{2},\\:y=x^{2}+6$$", "symbolab_question": "INTEGRAL_APPLICATION#area y=4x^{2},y=x^{2}+6" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integral Applications", "subTopic": "Area between curves", "default": "8\\sqrt{2}", "decimal": "11.31370…" }, "steps": { "type": "interim", "title": "The area between the curves $$y=4x^{2}\\:$$and $$y=x^{2}+6:{\\quad}8\\sqrt{2}$$", "steps": [ { "type": "definition", "title": "The area between curves definition", "text": "The area between curves is the area between a curve $$f\\left(x\\right)\\:$$and a curve $$g\\left(x\\right)\\:$$on an interval $$[a,\\:b]\\:$$given by<br/>$$A=\\int_{a}^{b}|f\\left(x\\right)-g\\left(x\\right)|dx$$" }, { "type": "interim", "title": "Apply the area formula:$${\\quad}\\int_{-\\sqrt{2}}^{\\sqrt{2}}\\left|4x^{2}-\\left(x^{2}+6\\right)\\right|dx$$", "steps": [ { "type": "step", "primary": "$$f\\left(x\\right)=4x^{2}$$", "secondary": [ "$$g\\left(x\\right)=x^{2}+6$$" ] }, { "type": "interim", "title": "Find intersection points:$${\\quad}x=\\sqrt{2},\\:x=-\\sqrt{2}$$", "steps": [ { "type": "step", "primary": "To find the intersection points solve $$f\\left(x\\right)=g\\left(x\\right)$$", "result": "4x^{2}=x^{2}+6" }, { "type": "interim", "title": "$$4x^{2}=x^{2}+6{\\quad:\\quad}x=\\sqrt{2},\\:x=-\\sqrt{2}$$", "input": "4x^{2}=x^{2}+6", "result": "x=\\sqrt{2},\\:x=-\\sqrt{2}", "steps": [ { "type": "interim", "title": "Move $$x^{2}\\:$$to the left side", "input": "4x^{2}=x^{2}+6", "result": "3x^{2}=6", "steps": [ { "type": "step", "primary": "Subtract $$x^{2}$$ from both sides", "result": "4x^{2}-x^{2}=x^{2}+6-x^{2}" }, { "type": "step", "primary": "Simplify", "result": "3x^{2}=6" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3x^{2}=6", "result": "x^{2}=2", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3x^{2}}{3}=\\frac{6}{3}" }, { "type": "step", "primary": "Simplify", "result": "x^{2}=2" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mH82w3phdmM14KufnEEtCOhsd01nKklADWWESC8hlR88BM0yvajEwZUfPEKsYNjoDojsf8lKRTEudrXmZFtoNiGhb2PWuGavmkmAcW9J/elbwVz5T7/Dyunz8G+H2e0kUsD+pIkOCIZyIU4q8atffPpdu80I8/0M2UIpmdiK+fxM52y9xQsIUCHttNibn1E6j/zwv/N01sJwA0CLJV/QZ0jUIy+hj9iWxfsoWUXffcyv7bEuL0hS9H7N+zuEXdPA1ISdTxIJU4Hy+c+iMHNQV61WOHyemdu4lkiXdkHmn+mudN00Zudf6MEHEXZQ/T2OmWxVBWFQ2bnWzaDkr1TK3DGzUx6L3ubZhb/zeksJ/Yx5HMze/KerEkoiDeO0X1kJj51HCRqdlolTDJhJwLVMLO+VSz2yw3ga51Ab5U4/cafK3wsVCblSmxuxnmJ9OSEUZGkgmoxoNlYnAOkblJbByx3tkbpEhAhHqWbUSSXvVLtFKk3fejFkyiOiq9iG9IkAybeXGnJZsE3KymoroUC4qgqsD5HFp2HnAUw4NpIuPmM=" } }, { "type": "step", "primary": "For $$x^{2}=f\\left(a\\right)$$ the solutions are $$x=\\sqrt{f\\left(a\\right)},\\:\\:-\\sqrt{f\\left(a\\right)}$$" }, { "type": "step", "result": "x=\\sqrt{2},\\:x=-\\sqrt{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } } ], "meta": { "interimType": "Find Intersection Title 0Eq" } }, { "type": "step", "primary": "Therefore" }, { "type": "step", "primary": "$$a=-\\sqrt{2},\\:b=\\sqrt{2}$$" }, { "type": "step", "result": "=\\int_{-\\sqrt{2}}^{\\sqrt{2}}\\left|4x^{2}-\\left(x^{2}+6\\right)\\right|dx" } ], "meta": { "interimType": "Apply Area Formula 0Eq" } }, { "type": "interim", "title": "Solve $$\\int_{-\\sqrt{2}}^{\\sqrt{2}}\\left|4x^{2}-\\left(x^{2}+6\\right)\\right|dx:{\\quad}8\\sqrt{2}$$", "input": "\\int_{-\\sqrt{2}}^{\\sqrt{2}}\\left|4x^{2}-\\left(x^{2}+6\\right)\\right|dx", "steps": [ { "type": "interim", "title": "Eliminate Absolutes", "result": "=\\int_{-\\sqrt{2}}^{\\sqrt{2}}-3x^{2}+6dx", "steps": [ { "type": "step", "primary": "Find the equivalent expressions to $$\\left|4x^{2}-\\left(x^{2}+6\\right)\\right|$$ at $$-\\sqrt{2}\\le\\:x\\le\\:\\sqrt{2}$$ without the absolutes" }, { "type": "step", "primary": "$$-\\sqrt{2}\\le\\:x\\le\\:\\sqrt{2}:{\\quad}-3x^{2}+6$$" }, { "type": "step", "result": "=\\int_{-\\sqrt{2}}^{\\sqrt{2}}-3x^{2}+6dx" } ], "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_{-\\sqrt{2}}^{\\sqrt{2}}3x^{2}dx+\\int_{-\\sqrt{2}}^{\\sqrt{2}}6dx" }, { "type": "interim", "title": "$$\\int_{-\\sqrt{2}}^{\\sqrt{2}}3x^{2}dx=4\\sqrt{2}$$", "input": "\\int_{-\\sqrt{2}}^{\\sqrt{2}}3x^{2}dx", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=3\\cdot\\:\\int_{-\\sqrt{2}}^{\\sqrt{2}}x^{2}dx" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int_{-\\sqrt{2}}^{\\sqrt{2}}x^{2}dx", "result": "=3[\\frac{x^{3}}{3}]_{-\\sqrt{2}}^{\\sqrt{2}}", "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}]_{-\\sqrt{2}}^{\\sqrt{2}}" }, { "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}]_{-\\sqrt{2}}^{\\sqrt{2}}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s75Rm4vRiKnW6efC3FQUj7HTH3EpEcp+IhEeUP9FH743cHI5S0StY1FdtOqqOPr0Te6dTGBOUIR1F3F0c+HOjNXwoCBaLRbuqMjNF1pfjz+TpiAE+4srYFw7NqQJs6tu72Gc2CCOKV4sfIGPRkz+HwQhLfvJvgdVfVsAJtY3akO39ZEt3ZXAiqUE0HIXrrrezJKpEiuPSdewDMVesjro3/3q81vfCB6Ypk1wEa+4zGRLR" } }, { "type": "interim", "title": "Compute the boundaries:$${\\quad}\\frac{4\\sqrt{2}}{3}$$", "input": "[\\frac{x^{3}}{3}]_{-\\sqrt{2}}^{\\sqrt{2}}", "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": "$$\\lim_{x\\to\\:-\\sqrt{2}+}\\left(\\frac{x^{3}}{3}\\right)=-\\frac{2\\sqrt{2}}{3}$$", "input": "\\lim_{x\\to\\:-\\sqrt{2}+}\\left(\\frac{x^{3}}{3}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=-\\sqrt{2}$$", "result": "=\\frac{\\left(-\\sqrt{2}\\right)^{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 $$\\frac{\\left(-\\sqrt{2}\\right)^{3}}{3}:{\\quad}-\\frac{2\\sqrt{2}}{3}$$", "input": "\\frac{\\left(-\\sqrt{2}\\right)^{3}}{3}", "result": "=-\\frac{2\\sqrt{2}}{3}", "steps": [ { "type": "interim", "title": "$$\\left(-\\sqrt{2}\\right)^{3}=-\\left(\\sqrt{2}\\right)^{3}$$", "input": "\\left(-\\sqrt{2}\\right)^{3}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=-a^{n},\\:$$if $$n$$ is odd", "secondary": [ "$$\\left(-\\sqrt{2}\\right)^{3}=-\\left(\\sqrt{2}\\right)^{3}$$" ], "result": "=-\\left(\\sqrt{2}\\right)^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CjG3phX13/UUPsxTXr6p3eT0JTwwJam8DFLG9sjVKUN5tMpJTBBccUWkSyvMe1Sp/khD5nsnWEIGwIqSHU5sDvEq6qw1uJhXBCvoTzamTfJGuI1WDbiNhHIdSl/p3qh8ykmQRYuUtVCKuKKnJrY2KbCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=\\frac{-\\left(\\sqrt{2}\\right)^{3}}{3}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{\\left(\\sqrt{2}\\right)^{3}}{3}" }, { "type": "interim", "title": "$$\\left(\\sqrt{2}\\right)^{3}=2\\sqrt{2}$$", "input": "\\left(\\sqrt{2}\\right)^{3}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(2^{\\frac{1}{2}}\\right)^{3}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=2^{\\frac{1}{2}\\cdot\\:3}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:3=\\frac{3}{2}$$", "input": "\\frac{1}{2}\\cdot\\:3", "result": "=2^{\\frac{3}{2}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:3}{2}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3=3$$", "result": "=\\frac{3}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8SPpFLppSOOOwknMsNOrv36rju+5Z51e/ZZSD3gRHwjBnpkZpWG0liIE8buj+sqXg2RLd2VwIqlBNByF6663sySF76Eydb/wnfqEDL8zJUMWNJdlTvBkWjQ+TKBgKD6qgLCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "$$2^{\\frac{3}{2}}=2\\sqrt{2}$$", "input": "2^{\\frac{3}{2}}", "result": "=2\\sqrt{2}", "steps": [ { "type": "step", "primary": "$$2^{\\frac{3}{2}}=2^{1+\\frac{1}{2}}$$", "result": "=2^{1+\\frac{1}{2}}" }, { "type": "step", "primary": "Apply exponent rule: $$x^{a+b}=x^{a}x^{b}$$", "result": "=2^{1}\\cdot\\:2^{\\frac{1}{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Refine", "result": "=2\\sqrt{2}" } ], "meta": { "interimType": "N/A" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tCsawgaxLQobh+JTWw95k3yRHuGw7+tM5METTDj6vVHUl62n0C+hBxjZBIPAUmHP4d/Uj9s/8iP228jznyr7FFv5MfuUt1xOquuYEoQ6XNJaDxuB6L7qNWw5+M9nJ/QRjwE87HTCWyAU3ypRroDMDQ==" } }, { "type": "step", "result": "=-\\frac{2\\sqrt{2}}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7eETGhp+onjWm8FoUrkXHtcs3Q6ZwptIM5Y793yU7dJ3ehkKrn0era9rz8TlL+x/vttdvQxZI3PlVepHWO3+UgrboyXPyoukNR3eN/+NbQnQ/y9DKGIPglJ+qMi9xDu2KaRI7GCp0HQz+zDw23axddEL0YFsw86QYOJ3rluNITLSnQVEaLYpoEHFkFyCn6I7KJLd1ohke2Wgml78++2zI0g==" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:\\sqrt{2}-}\\left(\\frac{x^{3}}{3}\\right)=\\frac{2\\sqrt{2}}{3}$$", "input": "\\lim_{x\\to\\:\\sqrt{2}-}\\left(\\frac{x^{3}}{3}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=\\sqrt{2}$$", "result": "=\\frac{\\left(\\sqrt{2}\\right)^{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": "$$\\left(\\sqrt{2}\\right)^{3}=2\\sqrt{2}$$", "input": "\\left(\\sqrt{2}\\right)^{3}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(2^{\\frac{1}{2}}\\right)^{3}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=2^{\\frac{1}{2}\\cdot\\:3}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:3=\\frac{3}{2}$$", "input": "\\frac{1}{2}\\cdot\\:3", "result": "=2^{\\frac{3}{2}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:3}{2}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3=3$$", "result": "=\\frac{3}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8SPpFLppSOOOwknMsNOrv36rju+5Z51e/ZZSD3gRHwjBnpkZpWG0liIE8buj+sqXg2RLd2VwIqlBNByF6663sySF76Eydb/wnfqEDL8zJUMWNJdlTvBkWjQ+TKBgKD6qgLCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "$$2^{\\frac{3}{2}}=2\\sqrt{2}$$", "input": "2^{\\frac{3}{2}}", "result": "=2\\sqrt{2}", "steps": [ { "type": "step", "primary": "$$2^{\\frac{3}{2}}=2^{1+\\frac{1}{2}}$$", "result": "=2^{1+\\frac{1}{2}}" }, { "type": "step", "primary": "Apply exponent rule: $$x^{a+b}=x^{a}x^{b}$$", "result": "=2^{1}\\cdot\\:2^{\\frac{1}{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Refine", "result": "=2\\sqrt{2}" } ], "meta": { "interimType": "N/A" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tCsawgaxLQobh+JTWw95k3yRHuGw7+tM5METTDj6vVHUl62n0C+hBxjZBIPAUmHP4d/Uj9s/8iP228jznyr7FFv5MfuUt1xOquuYEoQ6XNJaDxuB6L7qNWw5+M9nJ/QRjwE87HTCWyAU3ypRroDMDQ==" } }, { "type": "step", "result": "=\\frac{2\\sqrt{2}}{3}" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=\\frac{2\\sqrt{2}}{3}-\\left(-\\frac{2\\sqrt{2}}{3}\\right)" }, { "type": "interim", "title": "Simplify $$\\frac{2\\sqrt{2}}{3}-\\left(-\\frac{2\\sqrt{2}}{3}\\right):{\\quad}\\frac{4\\sqrt{2}}{3}$$", "input": "\\frac{2\\sqrt{2}}{3}-\\left(-\\frac{2\\sqrt{2}}{3}\\right)", "result": "=\\frac{4\\sqrt{2}}{3}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{2\\sqrt{2}}{3}+\\frac{2\\sqrt{2}}{3}" }, { "type": "step", "primary": "Add similar elements: $$\\frac{2\\sqrt{2}}{3}+\\frac{2\\sqrt{2}}{3}=2\\cdot\\:\\frac{2\\sqrt{2}}{3}$$", "result": "=2\\cdot\\:\\frac{2\\sqrt{2}}{3}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{2\\sqrt{2}\\cdot\\:2}{3}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{4\\sqrt{2}}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iA1jhOJsYg26KMh1oIvzcpsNzQPyCUAvDKXqafj5w4LnNZ4NIPSnUCAupMi4JKAWAJYpRu9XpYrd8NSAW2DdD/KxLrO04AooUAReaJjhZCan0RXq8/1t4lpPIZ7VjRieFJh4PokMRpRsSlg2/Mpicx4pgUWEah0lniZLlD4X0wt7wEILmCJxKqDXhc6wLMohEKq5N13cbB7cZUHHwBe0dtQ8Ok5Gs/3/xf5JwJWRK8xoVMIvGpWk/e1th+v9wUgY" } } ], "meta": { "interimType": "Integral Definite Limit Boundaries 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s75Rm4vRiKnW6efC3FQUj7HT7vx1Jiqs8NAHCayK7Ezvu/koYGDkQ4SWDRIfahfSErGPi7JPpD1bJizhEUxAn5aGkCZC+X826XfpfUefMS9IK7l00Ep3JjtFIt7PBKlMbNi9ceW7Wodb2p8vpx8u6SfF6pfF1z6umzUJTJvt+ojYZiLAByJF0Xur9AkPqhZibVt0/CjtCbVTqhBKA72P2l6M=" } }, { "type": "step", "result": "=3\\cdot\\:\\frac{4\\sqrt{2}}{3}" }, { "type": "step", "primary": "Simplify", "result": "=4\\sqrt{2}" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int_{-\\sqrt{2}}^{\\sqrt{2}}6dx=12\\sqrt{2}$$", "input": "\\int_{-\\sqrt{2}}^{\\sqrt{2}}6dx", "steps": [ { "type": "step", "primary": "Integral of a constant: $$\\int{a}dx=ax$$", "result": "=[6x]_{-\\sqrt{2}}^{\\sqrt{2}}" }, { "type": "interim", "title": "Compute the boundaries:$${\\quad}12\\sqrt{2}$$", "input": "[6x]_{-\\sqrt{2}}^{\\sqrt{2}}", "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": "$$\\lim_{x\\to\\:-\\sqrt{2}+}\\left(6x\\right)=-6\\sqrt{2}$$", "input": "\\lim_{x\\to\\:-\\sqrt{2}+}\\left(6x\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=-\\sqrt{2}$$", "result": "=6\\left(-\\sqrt{2}\\right)", "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": "=-6\\sqrt{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:\\sqrt{2}-}\\left(6x\\right)=6\\sqrt{2}$$", "input": "\\lim_{x\\to\\:\\sqrt{2}-}\\left(6x\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=\\sqrt{2}$$", "result": "=6\\sqrt{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 $$\\lim_{x\\to{a}}g\\left(x\\right)\\neq0$$" } } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=6\\sqrt{2}-\\left(-6\\sqrt{2}\\right)" }, { "type": "step", "primary": "Simplify", "result": "=12\\sqrt{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "interimType": "Integral Definite Limit Boundaries 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s75Rm4vRiKnW6efC3FQUj7HRqZVthvahsjAFF1K6uWfRPmD4eAZwiOCRbsqaslmqEkHTGoPaEPibF3nUFwMnE7TJBZEPoY5PC1Y62C4s1scueMtz236Fjo1vDzPH+XuzIiIuJ9L+ej+UqBQHrKiwMlGMGpyDzFP/1AzB2560RSVH86r7lU85FeN8KANEd7x+svQ==" } }, { "type": "step", "result": "=12\\sqrt{2}" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=-4\\sqrt{2}+12\\sqrt{2}" }, { "type": "step", "primary": "Simplify", "result": "=8\\sqrt{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The area is:", "result": "=8\\sqrt{2}" } ] }, "plot_output": { "meta": { "plotInfo": { "variable": "y", "plotRequest": "yes" }, "showViewLarger": true } } }