The volume of the solid obtained by rotating the region bounded by the graphs y = -1, y = 0, x = 1, and x = 3 about the axis y = 8 using cylindrical shells is 48π.
Identify the region to be rotated and the axis of rotation. In this case, the region is bounded by the graphs y = -1, y = 0, x = 1 and x = 3, and the axis of rotation is y = 8.
Draw a representative cylindrical shell with radius r, height h and thickness dx. The radius is the distance from the axis of rotation to the shell, so r = 8 - x. The height is the length of the shell along the axis of rotation, so h = 1 - (-1) = 2. The thickness is the width of the shell along the x-axis, so dx is a small change in x.
Write the formula for the volume of a single shell. The volume is the product of the circumference, height and thickness of the shell, so V = 2πrhdx = 2π(8 - x)(2)dx.
Write the integral for the total volume of the solid. The integral is the sum of the volumes of all the shells from x = 1 to x = 3, so V = ∫13 2π(8 - x)(2)dx.
Evaluate the integral using the power rule and the fundamental theorem of calculus. V = 2π ∫13 (16 - 2x)dx = 2π [16x - x2]13 = 2π [(16(3) - (3)2) - (16(1) - (1)2)] = 2π (39 - 15) = 48π.