Answer to Question #185969 in Calculus for jmcabanes

Question #185969

Rotate the region bounded by x

=

(

y

2

)

2

x=(y−2)2

, the x

x

-axis and the y

y

-axis about the x

x

-axis.


1
Expert's answer
2021-05-07T09:26:51-0400

Given, the bounded region


Volume generated by the bounded region about the x-axis=2πaby2(x)dx where x varies from a to b.=2π04(2x)2dx=2π04(4+x4x)dx=2π[4x+x228x323]04=2π(83)=163π units32 \pi \int_{a}^{b}y^2(x)dx\space where \space x \space varies \space from \space a\space to\space b.\newline =2 \pi \int_{0}^{4}(2- \sqrt{x})^2dx\newline =2 \pi \int_{0}^{4}(4+x-4 \sqrt{x})dx\newline =2 \pi [4x+\frac{x^2}{2}-8 \frac{x^{\frac{3}{2}}}{3}]_{0}^{4}\newline =2\pi (\frac{8}{3})\newline =\frac{16}{3} \pi\space units^3

Thus, the required volume is 163π units3\frac{16}{3} \pi\space units^3.


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