let R be the region bounded by xy3 x27 and y0 Compute the vo

Solution

Although not mathematically rigorous, you can think of the volume as the sum of the volumes of an infinite number of infinitely thin circular discs. The radius of each disc is a line parallel to the x-axis between the graphs of y = x^3 and x = 2. The area of each disc = pi * r^2 where r is the length of that line. If y = x^3 then x = y^(1/3) so r = 2 - y^(1/3) and each disc has an area of pi*[2 - y^(1/3)]^2 If we solve y = x^3 and x = 2 together, we get y = 2^3 = 8. So y = x^3 and x = 2 intersect at (2, 8). So the y values of the discs go from y = 1 to 8. The volume = integral (y = 1 ---> 8) pi * [2 - y^(1/3)]^2 dy = pi integral (y = 1 ---> 8) [4 - 4y^(1/3) + y^(2/3)] dy = pi [4y - 3y^(4/3) + (3/5)y^(5/3)] | (y = 1 ---> 8) = pi{[32 - 48 + (3/5)*32] - [4 - 3 +(3/5)]} = pi (16/5 - 8/5) = 8pi/5