# Let V be the volume of the solid obtained by rotating the region bounded by the following curves...

## Question:

Let {eq}V {/eq} be the volume of the solid obtained by rotating the region bounded by the following curves about the {eq}y {/eq}-axis.

{eq}y = \sqrt{x} {/eq}, {eq}y = x^2 {/eq}

A) Find {eq}V {/eq} by slicing.

B) Find {eq}V {/eq} by cylindrical shells.

## Finding the Volume of a Solid of Revolution by the Washer Method and the Shell Method:

The Washer Method says that the volume of the solid formed by rotating the area between the curves *x* = *f*(*y*) and *x* = *g*(*y*) and the lines *y* = *a* and *y* = *b* about the *y*-axis is given by

{eq}V=\pi \int_{a}^{b}|f^{2}(y)-g^{2}(y)|dy {/eq}

The Shell Method says that the volume of the solid formed by rotating the area between the curves *y* = *f*(*x*) and *y* = *g*(*x*) and the lines *x* = *a* and *x* = *b* about the *y*-axis is given by

{eq}V=2\pi \int_{a}^{b} x|f(x) - g(x)| dx {/eq}

## Answer and Explanation:

First we want to find the intersections of the two curves by solving the system of equations

{eq}y = \sqrt{x} \\ y = x^{2}. {/eq}

Substituting one...

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