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?

Verified step by step guidance

1. 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 \([0, \pi]\).
2. 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 \(\sin(x)\).
3. 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.
4. Set up both integrals explicitly: \(V_x = \pi \int_0^{\pi} \sin^2(x) \, dx\) \(V_y = 2\pi \int_0^{\pi} x \sin(x) \, dx\)
5. 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.

Key Concepts

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.