Solid of Revolution about Horizontal Line (Washer)

Question

Let the region R be the area enclosed by the function f(x)=e^{x}+2 and g(x)=4x+3. Find the volume of the solid generated when the region R is revolved about the line y=2. You may use a calculator and round to the nearest thousandth.

Answer 1

V = ∫_a^bπ|(f(x)-d)²-(g(x)-d)²|dx, where d is the line of rotation

The hardest part is determining [a,b]. If you plot the two functions with a plotting tool, you can determine the interval to be the intersection of the two functions.

[a,b]=[0, 2.337]. Or you can use newtons method to solve for x:

h(x) = e^x+2-(4x+3)

which gives 2.337 after 5 iterations.

V=π ∫_a^b|e^2x-(4x+1)²|dx = |e^2x/2 - 16x³/3-4x²-x|_a^b