Surface area of revolution
Applications of integration · Mathematics
Study notes
Q: Find the surface area when y = sqrt(x), 1 to 4, is rotated about the x-axis. dy/dx = 1/(2 sqrt(x)). 1 + (dy/dx)^2 = 1 + 1/(4x) = (4x+1)/(4x). ds = sqrt((4x+1)/(4x)) dx = sqrt(4x+1)/(2 sqrt(x)) dx. S = 2 pi integral 1 to 4 of sqrt(x) times sqrt(4x+1)/(2 sqrt(x)) dx = pi integral 1 to 4 of sqrt(4x+1) dx. Let u = 4x+1: pi times [u^(3/2)/6] 5 to 17 = (pi/6)(17 sqrt(17) - 5 sqrt(5)).