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Find the volume of the solid obtained by rotating the region bounded by the given curves about...

Question:

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

{eq}y = 40x - 5x^2, \; y = 0 \, ; {/eq} about the {eq}y {/eq}-axis

Answer and Explanation:

{eq}\text{Finding the limits of integration:}\\ 40x-5x^2=0\\ 5x(8-x)=0\\ x=0~x=8\\ \text{We will use the shell method. We will find the radius and the height of the shell.}\\ r=x\\ h=40x-5x^2\\ \text{The formula for the volume:}\\ V=2\pi\int_{a}^{b}rhdx\\ \text{Applying:}\\ V=2\pi\int_{0}^{8}x(40x-5x^2)dx\\ =2\pi\int_{0}^{8}\left (40x^2-5x^3 \right )dx\\ \text{Applying the formula:}\\ =2\pi\left [\frac{40x^3}{3}-\frac{5x^4}{4} \right ]_{0}^{8}\\ =\frac{10240\pi}{3}\\ \text{is the volume.} {/eq}


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