Mathematics, 11.06.2020 13:57 samarkanduzbek
Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region bounded by y 1 3 ex2 /3 , y 0, x 0, and x 3 about the y-axis. Round your answer to three decimal places.
Answers: 1
![At,\\\\ x=0, y=0 \ and \ At, x=3, y=-\sqrt[3]{9}](/tpl/images/0682/8661/1f3fb.png)
![\int_{0}^{V}dV=2\sqrt{\frac{\pi}{3}}\int_{0}^{3}x^{\frac{4}{3}}e^{-x^{\frac{2}{3}}}x^{\frac{-1}{3}}dx=2\sqrt{\frac{\pi}{3}}\int_{0}^{-\sqrt[3]{9}}y^{2}e^{y}(\frac{-3}{2})dy](/tpl/images/0682/8661/61cc3.png)
![V=-\sqrt{3\pi}\int_{0}^{-\sqrt[3]{9}}y^{2}e^{y}dy](/tpl/images/0682/8661/1fb00.png)
![I=\int_{0}^{-\sqrt[3]{9}}y^{2}e^{y}dy](/tpl/images/0682/8661/40158.png)
![\int udv=uv-\int vdu = y^2e^y-2[ye^y-e^y]=e^y[y^2-2y+2]](/tpl/images/0682/8661/b58af.png)
![I=[e^y(y^2-2y+2)]_{0}^{-\sqrt[3]{9}}\\\\=e^{-2.0802}[(2.0802)^2+2(2.0802)+2]-e^{0}[0-0+2]\\\\\frac{(4.3272+4.1604+2)}{8.0061}-2\\\\=\frac{10.4876}{8.0061}-2\\\\=1.3099-2\\\\=-0.6901](/tpl/images/0682/8661/4d323.png)
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Use the shell method to set up and evaluate the integral that gives the volume of the solid generate...
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