101. Comparing volumes Let R be the region bounded by the graph of y = sin(x) and the x-axis on the interval [0, π]. Which is greater, the volume of the solid generated when R is revolved about the x-axis or 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 \(V_y = 2\pi \int_0^{\pi} x \sin(x) \, dx\), where \(x\) is the radius of the shell and \(\sin(x)\) is the height.

Set up both integrals explicitly: \(V_x = \pi \int_0^{\pi} \sin^2(x) \, dx\) \(V_y = 2\pi \int_0^{\pi} x \sin(x) \, dx\)

Evaluate both integrals separately (using appropriate integration techniques such as power-reduction for \(\sin^2(x)\) and integration by parts for \(x \sin(x)\)), then compare the two volumes to determine which is greater.

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Volume of Solids of Revolution

This concept involves finding the volume of a 3D solid formed by rotating a 2D region around an axis. Common methods include the disk/washer method for rotation about the x-axis and the shell method for rotation 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 rotation, creating circular disks or washers. The volume is found by integrating the area of these cross-sectional disks along the interval, typically using the formula π∫[f(x)]² dx.

Shell Method

This method is useful for revolving a region around the y-axis. It involves slicing the region parallel to the axis of rotation, forming cylindrical shells. The volume is calculated by integrating the lateral surface area of these shells, using the formula 2π∫(radius)(height) dx.