Tentukan volume benda putar yang terjadi bila daerah yang dibatasi kurva y = 8 - 2x, garis x = 1, dan x = 3 diputar mengelilingi sumbu x!
izharulhaq78
![\displaystyle L=\pi\int\limits^{x_2}_{x_1}y^2\,dx\\L=\pi\int\limits^3_1(8-2x)^2\,dx\\\\\text{misal :}\\8-2x=u\\-2\,dx=du\\\\L=\pi\int\limits^3_1u^2\cdot\left(-\frac12\,du\right)\\L=-\frac12\pi\int\limits^3_1u^2\,du\\L=\left-\frac16\pi u^3\right|^3_1\\L=\left-\frac16\pi(8-2x)^3\right|^3_1\\L=-\frac16\pi\left[(8-2(3))^3-(8-2(1))^3\right]\\L=-\frac16\pi\left[8-216\right]\\L=-\frac16\pi[-208]\\\boxed{\boxed{L=\frac{104}3\pi\,\text{sV}}}](https://tex.z-dn.net/?f=%5Cdisplaystyle+L%3D%5Cpi%5Cint%5Climits%5E%7Bx_2%7D_%7Bx_1%7Dy%5E2%5C%2Cdx%5C%5CL%3D%5Cpi%5Cint%5Climits%5E3_1%288-2x%29%5E2%5C%2Cdx%5C%5C%5C%5C%5Ctext%7Bmisal+%3A%7D%5C%5C8-2x%3Du%5C%5C-2%5C%2Cdx%3Ddu%5C%5C%5C%5CL%3D%5Cpi%5Cint%5Climits%5E3_1u%5E2%5Ccdot%5Cleft%28-%5Cfrac12%5C%2Cdu%5Cright%29%5C%5CL%3D-%5Cfrac12%5Cpi%5Cint%5Climits%5E3_1u%5E2%5C%2Cdu%5C%5CL%3D%5Cleft-%5Cfrac16%5Cpi+u%5E3%5Cright%7C%5E3_1%5C%5CL%3D%5Cleft-%5Cfrac16%5Cpi%288-2x%29%5E3%5Cright%7C%5E3_1%5C%5CL%3D-%5Cfrac16%5Cpi%5Cleft%5B%288-2%283%29%29%5E3-%288-2%281%29%29%5E3%5Cright%5D%5C%5CL%3D-%5Cfrac16%5Cpi%5Cleft%5B8-216%5Cright%5D%5C%5CL%3D-%5Cfrac16%5Cpi%5B-208%5D%5C%5C%5Cboxed%7B%5Cboxed%7BL%3D%5Cfrac%7B104%7D3%5Cpi%5C%2C%5Ctext%7BsV%7D%7D%7D "\displaystyle L=\pi\int\limits^{x_2}_{x_1}y^2\,dx\\L=\pi\int\limits^3_1(8-2x)^2\,dx\\\\\text{misal :}\\8-2x=u\\-2\,dx=du\\\\L=\pi\int\limits^3_1u^2\cdot\left(-\frac12\,du\right)\\L=-\frac12\pi\int\limits^3_1u^2\,du\\L=\left-\frac16\pi u^3\right|^3_1\\L=\left-\frac16\pi(8-2x)^3\right|^3_1\\L=-\frac16\pi\left[(8-2(3))^3-(8-2(1))^3\right]\\L=-\frac16\pi\left[8-216\right]\\L=-\frac16\pi[-208]\\\boxed{\boxed{L=\frac{104}3\pi\,\text{sV}}}")
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