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Find the volume of the solid generated by rotating about the y axis region bounded by the given...

Question:

Find the volume of the solid generated by rotating about the y axis region bounded by the given equations.

{eq}\displaystyle x = y^ {\frac{3}{2}},\ x = 0\ \text{and}\ y = 8 {/eq}.

Volume:

We have a function which is a power function with fractional index. We have to find the volume. We will find it with the help of the disk method. We will find the radius and then apply the formula.

Answer and Explanation:

{eq}\displaystyle x = y^ {\frac{3}{2}} {/eq}

We will first find the limits of integration.

{eq}y=0~y=8 {/eq}

We will apply the disk method. We will find the radius which is the distance of the function from the axis of rotation:

{eq}displaystyle r=y^{\frac{3}{2}} {/eq}

The formula for the volume:

{eq}V=\pi\int_{a}^{b}r^2dy\\ V=\pi\int_{0}^{8}y^3dy {/eq}

We will apply the integral formula:

{eq}displaystyle V=\pi \left [\frac{y^4}{4} \right ]_{0}^{8}\\ \displaystyle V=\pi\left [\frac{8^4}{4}-0 \right ] \\ V=1024\pi {/eq}


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