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

Question:

Find the volume {eq}V {/eq} of the solid obtained by rotating the region bounded by the given curves about the specified axis.

{eq}x = 6 \sqrt{3y}, \; x = 0, \; y = 3 \, ; \; {/eq} about the {eq}y {/eq}-axis

Answer and Explanation:

{eq}\text{Finding the limits:}\\ x=6\sqrt{3y}\\ x=0\\ 6\sqrt{3y}=0\\ y=0\\ \text{The other limit is given in the question:}\\ y=3\\ \text{We will use the disk method.}\\ \text{We will find the radius of the disk which is the distance of the function form the axis of rotation:}\\ r=6\sqrt{3y}\\ \text{The formula for the disk method:}\\ V=\pi\int_{a}^{b}r^2dy\\ \text{Applying}\\ V=\pi\int_{0}^{3}108ydy\\ \text{Applying the formula:}\\ V=\pi\left [54y^2 \right ]_{0}^{3}\\ =486\pi\\ \text{is the volume.} {/eq}


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