Chapter Review Exercise Q12

Chapter Review Exercise Q12

Postby chriskwee on Wed Feb 25, 2009 3:37 am

\[
\begin{gathered}
y = 8 - x^2 \hfill \\
when{\text{ }}x = 2 \hfill \\
y = 8 - 2^2 \hfill \\
y = 4 \hfill \\
{\text{For }}y = 8 - x^2 \hfill \\
x^2 = 8 - y \hfill \\
{\text{Volume of the shaded region about the y - axis}} \hfill \\
{\text{ = Volume of cylinder - }}\int_{\text{4}}^{\text{8}} {\pi {\text{x}}^{\text{2}} dy} \hfill \\
= \pi (2)^2 (4) - \int_{\text{4}}^{\text{8}} {\pi (8 - y)dy} \hfill \\
= 16\pi - \pi \left[ {8y - \frac{{y^2 }}
{2}} \right]_4 ^8 \hfill \\
= 16\pi - \pi \left[ {8(8) - \frac{{8^2 }}
{2} - \left( {8(4) - \frac{{4^2 }}
{2}} \right)} \right] \hfill \\
= 16\pi - \pi \left[ {64 - 32 - \left( {32 - 8} \right)} \right] \hfill \\
= 16\pi - \pi \left[ {32 - 24} \right] \hfill \\
= 16\pi - 8\pi \hfill \\
= 8\pi {\text{ cubit units}} \hfill \\
\end{gathered}
\]
chriskwee
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