Question

question 9
4 pts
the integral ( v = pi int_{0}^{pi} left( sqrt{sin x}
ight)^2 dx ) represents the volume of a solid. choose the description that best describes the solid.

• the integral describes the volume of the solid obtained by rotating the region between ( 0 leq y leq pi ) and ( 0 leq x leq sqrt{sin y} ) about the ( x )-axis.
• no correct answer choice is given.
• the integral describes the volume of the solid obtained by rotating the region between ( 0 leq x leq pi ) and ( 0 leq y leq sqrt{sin x} ) about the ( x )-axis.
• the integral describes the volume of the solid obtained by rotating the region between ( -pi leq x leq 0 ) and ( 0 leq y leq sqrt{sin^{-1} x} ) about the ( x )-axis.

Explanation:

Step1: Recall Disk Method Formula

The disk method for volume when rotating about the \(x\)-axis is \(V=\pi\int_{a}^{b}[r(x)]^{2}dx\), where \(r(x)\) is the radius (distance from \(x\)-axis to the curve \(y = r(x)\)) and \(a,b\) are the limits of integration.

Step2: Analyze Given Integral

Given \(V=\pi\int_{0}^{\pi}(\sqrt{\sin x})^{2}dx\). Comparing with disk method, \(r(x)=\sqrt{\sin x}\), limits \(a = 0\), \(b=\pi\). This means the region is between \(x = 0\) to \(x=\pi\), and \(y\) ranges from \(0\) to \(y=\sqrt{\sin x}\) (since \(r(x)\) is the \(y\)-value, so \(0\leq y\leq\sqrt{\sin x}\)), rotated about the \(x\)-axis.

Step3: Evaluate Options

• First option: Region is in terms of \(y\) ( \(0\leq y\leq\pi\), \(0\leq x\leq\sqrt{\sin y}\)) and rotated about \(x\)-axis. But our integral is in terms of \(x\), so this is incorrect.
• Third option: Matches our analysis (region \(0\leq x\leq\pi\), \(0\leq y\leq\sqrt{\sin x}\) rotated about \(x\)-axis).
• Fourth option: Region is \( -\pi\leq x\leq0\) and \(y\) in terms of \(\sin^{-1}x\), which doesn't match the integral's limits and function.

Answer:

The integral describes the volume of the solid obtained by rotating the region between \(0 \leq x \leq \pi\) and \(0 \leq y \leq \sqrt{\sin x}\) about the \(x\)-axis. (The option corresponding to this description)