How to find the volume of a solid revolving around a line?

Question

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line y = 7.

y= sqrt(4-x)

x=0

y=0

Answer 1

V = π∫(f(x)-d)² - (g(x) -d)²dx Integrated from a to b

f(x) = √(4-x) and g(x) = 0

Solving for x when f(x) = 0, x = 4

a = 0, b = 4, d = 7

V = π∫(|√(4-x)-7)²-7²|dx from 0 to 4

=π∫|4-x-14(4-x)^1/2|dx = π|[4x -x²/2 +14(2/3)(4-x)^3/2]| Evaluate at 4 and 0

= π|[16-8 - 0 - 0 +0 -14(2/3)8| = π|(8 - 224/3| = 200π/3