Applications to Geometry

Question

Consider the region bounded by y = e^x, the x-axis, and the lines x - 0 and x = 6. Find the volume of the solid obtained by rotating the region about the horizontal line y = -3.

Round your answer to two decimal places.

V =

Answer 1

If you sketch this out, you'll see that the function y = e^x rotated around the horizontal line y = -3 makes a shape with cross sections that are washers. You can determine the cross-sectional area of some random washer to be π(r_o² - r_i²) where r_i is the inside radius which is 4 units (3 + e⁰ = 4) and r_o is the outside radius which is "3 + e^x".

You can then integrate these between x = zero and x = 6.