Question

For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis.

y=1/x+1,x=0 and x=2

Answer 1

To find the volume when the region bounded by the curves y=1/x+1, x=0, and x=2 is rotated around the y-axis, we can use the method of cylindrical shells.

Step-by-step explanation:

To find the volume when the region bounded by the curves y=1/x+1, x=0, and x=2 is rotated around the y-axis, we can use the method of cylindrical shells.

1. First, we need to draw and visualize the region bounded by the curves. The curve y=1/x+1 is a hyperbola that is shifted upward by 1 unit.
2. Next, we need to setup the integral to calculate the volume. The general formula for volume using cylindrical shells is V = ∫ 2πx f(x) dx, where f(x) represents the height of the shell at each value of x.
3. In this case, the height of the shell is given by y=1/x+1. Substituting this into the formula, we have V = ∫ 2πx (1/x+1) dx.
4. Simplifying the integral, V = ∫ 2π (1 + 1/x) dx = 2π(x + ln|x|) + C.

Therefore, the volume of the region when rotated around the y-axis is given by 2π(x + ln|x|) evaluated from x=0 to x=2.