69. Comparing volumes Let R be the region bounded by y = sin x and the x-axis on the interval [0, π]. Which is greater, the volume when R is revolved about the x-axis, or the volume when R is revolved about the y-axis?

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First, identify the region R bounded by the curve \(y = \sin x\) and the x-axis on the interval \([0, \pi]\). This means the region lies between \(y = 0\) and \(y = \sin x\) for \(x\) in \([0, \pi]\).

To find the volume when R is revolved about the x-axis, use the disk method. The volume \(V_x\) is given by the integral: \(V_x = \pi \int_0^{\pi} (\sin x)^2 \, dx\) because the radius of each disk is \(\sin x\).

To find the volume when R is revolved about the y-axis, use the shell method. The volume \(V_y\) is given by the integral: \(V_y = 2\pi \int_0^{\pi} x \cdot \sin x \, dx\) where \(x\) is the radius of the shell and \(\sin x\) is the height.

Set up both integrals explicitly and prepare to evaluate them (though we won't compute the final values here). This involves recognizing the integrals: \(\int_0^{\pi} (\sin x)^2 \, dx\) and \(\int_0^{\pi} x \sin x \, dx\).

After evaluating both integrals, compare the numerical values of \(V_x\) and \(V_y\) to determine which volume is greater.

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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 generated when a region is revolved around an axis. Common methods include the disk/washer method for revolution about the x-axis and the shell method for revolution about the y-axis. Understanding these methods helps set up the correct integrals for volume calculation.

Disk/Washer Method

Used when revolving a region around the x-axis, this method slices the solid perpendicular to the axis of revolution, creating circular disks or washers. The volume is found by integrating the area of these cross-sectional disks along the interval of interest.

Shell Method

This method is useful for revolving a region around the y-axis. It involves slicing the region parallel to the axis of revolution, forming cylindrical shells. The volume is calculated by integrating the lateral surface area of these shells over the given interval.