Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.
y=sin xon [0,π] and y=0 ; about the x-axis (Hint: Recall that sin^2 x=1 − cos2x / 2.

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Identify the region R bounded by the curves y = sin x and y = 0 on the interval [0, \(\pi\)]. This region lies between the curve y = sin x and the x-axis from x = 0 to x = \(\pi\).

Since the region is revolved about the x-axis, use the disk method to find the volume. The volume V is given by the integral formula: \(V = \pi \int_0^{\pi} [f(x)]^2 \, dx\) where \(f(x) = \sin x\) is the radius of the disk at position x.

Substitute \(f(x) = \sin x\) into the volume integral: \(V = \pi \int_0^{\pi} (\sin x)^2 \, dx\).

Use the given trigonometric identity to simplify the integrand: \(\sin^2 x = \frac{1 - \cos 2x}{2}\) Rewrite the integral as: \(V = \pi \int_0^{\pi} \frac{1 - \cos 2x}{2} \, dx\).

Split the integral into two simpler integrals and set up the expression for evaluation: \(V = \frac{\pi}{2} \int_0^{\pi} (1 - \cos 2x) \, dx = \frac{\pi}{2} \left[ \int_0^{\pi} 1 \, dx - \int_0^{\pi} \cos 2x \, dx \right]\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Volume of Solids of Revolution

This concept involves finding the volume of a 3D solid formed by rotating a 2D region around an axis. The disk or washer method is commonly used, where cross-sectional areas perpendicular to the axis of rotation are integrated over the interval. For rotation about the x-axis, the volume is found by integrating π times the square of the function representing the radius.

Trigonometric Identities

Trigonometric identities simplify expressions involving trigonometric functions. In this problem, the identity sin²x = (1 - cos 2x)/2 helps transform the integrand into a more manageable form for integration. Using such identities is essential to evaluate integrals involving powers of sine or cosine.

Definite Integration over an Interval

Definite integration calculates the exact area under a curve between two points, here from 0 to π. It is used to sum the infinitesimal volumes of disks or washers to find the total volume. Understanding how to set up and evaluate definite integrals is crucial for solving volume problems in calculus.

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.y=sin xon [0,π] and y=0 ; about the x-axis (Hint: Recall that sin^2 x=1 − cos2x / 2.