The region under the curve y=sin ^2 x from 0 to π is rotated about the x -axis.

step 1 We want to find a volume generated by this function from 0 to pi rotated about the x -assist. So

The region under the curve y=sin ^2 x from 0 to π...

$\begin{array}{l}{\text { The region under the curve } y=\sin ^{2} x \text { from } 0 \text { to } \pi \text { is rotated }} \\ {\text { about the } x \text { -axis. Find the volume of the resulting solid. }}\end{array}$

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Key Concepts

Solids of Revolution

This concept involves generating a three?dimensional solid by rotating a two?dimensional region around an axis. It is used to calculate volumes of objects that have rotational symmetry by integrating cross-sectional areas along the axis of revolution.

Disk Method

The disk method is a technique for finding the volume of a solid of revolution by slicing the solid perpendicular to the axis of rotation. Each slice is approximated by a disk whose volume is determined by its radius (given by the value of the function) and thickness, and the total volume is found by integrating these disk volumes over the interval.

Trigonometric Power-Reduction Identity

This identity is used to simplify integrals involving squared trigonometric functions. For example, rewriting sin²(x) as (1 – cos(2x))/2 simplifies the integration process by transforming a squared sine function into a form involving a cosine function with a double angle, which can be integrated more directly.

Definite Integration

Definite integration is a fundamental concept in calculus used to compute the exact accumulation of quantities over a specific interval. In the context of volume calculations, it sums up the contributions of infinitesimally thin slices (disks) across the given bounds to yield the total volume of the solid.

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