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| | [[File:Desmos graph 6.1.12.png|right|350px|]] |
| <math> | | <math> |
| \begin{align} | | \begin{align} |
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| <math>\int_{0}^{2} \left[(4x-x^2) - (x^2)^2\right] dx </math> | | <math>\int_{0}^{2} \left[(4x-x^2) - (x^2)\right] dx</math> |
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| <math> | | <math> |
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| <math>\int_{0}^{2} \left[(4x-x^2) - (x^2)^2\right]dx = \int_{-3}^{-2}\left((x^2)-(8-x^2)\right)dx + \int_{-2}^{2} \left((8-x^2) - (x^2)\right)dx + \int_{2}^{3}\left((x^2)-(8-x^2)\right)dx = \frac{14}{3} + \frac{64}{3} + \frac{14}{3} = \frac{92}{3}</math> | | <math>\int_{0}^{2} \left[(4x-x^2) - (x^2)\right]dx = \int_{0}^{2} (4x)dx</math> |
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| <math> | | <math> |
| \begin{align} | | \begin{align} |
| \int_{-3}^{-2}\left((x^2)-(8-x^2)\right)dx &= \int_{-3}^{-2}\left(2x^2-8)\right)dx \\[2ex]
| | &= \left[\frac{4x^2}{2}\right]\Bigg|_{0}^{2} \\[2ex] |
| | | &= \left[2x^2\right]\Bigg|_{0}^{2} \\[2ex] |
| &= \left[\frac{2x^3}{3}-8x\right]\Bigg|_{-3}^{-2} \\[2ex] | | &= \left[2(2)^2\right]-\left[2(0)^2\right] = 8-0 \\[2ex] |
| | | &= 8 |
| &= \left[\frac{2(-2)^3}{3}-8(-2)\right]-\left[\frac{2(-3)^3}{3}-8(-3)\right] \\[2ex] | |
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| &= \left[\frac{-16}{3}+16\right]-\left[\frac{-54}{3}+24\right] = \frac{38}{3}-8 \\[2ex]
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| &= \frac{14}{3}
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| \end{align}
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| </math>
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| <math>
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| \begin{align}
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| \int_{-2}^{2} \left((8-x^2) - (x^2)\right)dx &= \int_{-2}^{2}\left(8-2x^2\right)dx \\[2ex]
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| &= \left[8x-\frac{2x^3}{3}\right]\Bigg|_{-2}^{2} \\[2ex]
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| &= \left[8(2)-\frac{2(2)^3}{3}\right] - \left[8(-2)-\frac{2(-2)^3}{3}\right] \\[2ex] | |
| &= \left[16-\frac{16}{3}\right]-\left[-16+\frac{16}{3}\right] = 32-\frac{32}{3} \\[2ex]
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| &= \frac{64}{3} | |
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| \end{align} | | \end{align} |
| </math> | | </math> |
Latest revision as of 23:30, 22 September 2022
![{\displaystyle \int _{0}^{2}\left[(4x-x^{2})-(x^{2})\right]dx}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/6d1ebe4106c00946ea1b68c9603574e2f1fdbcec)
![{\displaystyle \int _{0}^{2}\left[(4x-x^{2})-(x^{2})\right]dx=\int _{0}^{2}(4x)dx}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/0406095dc5c7b6b0a0ae7a38f759508615fe8bb2)
![{\displaystyle {\begin{aligned}&=\left[{\frac {4x^{2}}{2}}\right]{\Bigg |}_{0}^{2}\\[2ex]&=\left[2x^{2}\right]{\Bigg |}_{0}^{2}\\[2ex]&=\left[2(2)^{2}\right]-\left[2(0)^{2}\right]=8-0\\[2ex]&=8\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/f4fe3a4714b1b9304b6b82d1c41558f4831928be)
