Given, the bounded region
Volume generated by the bounded region about the x-axis=2 π ∫ a b y 2 ( x ) d x w h e r e x v a r i e s f r o m a t o b . = 2 π ∫ 0 4 ( 2 − x ) 2 d x = 2 π ∫ 0 4 ( 4 + x − 4 x ) d x = 2 π [ 4 x + x 2 2 − 8 x 3 2 3 ] 0 4 = 2 π ( 8 3 ) = 16 3 π u n i t s 3 2 \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 2 π ∫ a b y 2 ( x ) d x w h ere x v a r i es f ro m a t o b . = 2 π ∫ 0 4 ( 2 − x ) 2 d x = 2 π ∫ 0 4 ( 4 + x − 4 x ) d x = 2 π [ 4 x + 2 x 2 − 8 3 x 2 3 ] 0 4 = 2 π ( 3 8 ) = 3 16 π u ni t s 3
Thus, the required volume is 16 3 π u n i t s 3 \frac{16}{3} \pi\space units^3 3 16 π u ni t s 3 .
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