Question

Find the volume V of the solid obtained by rotating the region bounded by the curves y = 5e⁽⁻ˣ⁾, y = 5, x = 4 about the line y = 10.

Answer 1

To find the volume V of the solid obtained by rotating the region bounded by the curves y = 5e^-x, y = 5, x = 4 about the line y = 10, we can use the method of cylindrical shells.

Step-by-step explanation:

To find the volume V of the solid obtained by rotating the region bounded by the curves y = 5e-x, y = 5, x = 4 about the line y = 10, we can use the method of cylindrical shells.

First, we need to find the limits of integration, which are x = 0 and x = 4. Next, we find the height of each shell, which is the difference between the line y = 10 and the curve y = 5e-x. This gives us the height h = 10 - 5e-x.

The radius of each shell is the distance from the line y = 10 to the x-axis, which is 10 - y = 10 - 5e-x. Now, we can set up the integral:

V = 2π ∫[0,4] (10 - 5e-x)(10 - 5e-x) dx

Solving this integral will give us the volume of the solid.