How do you find the volume of the solid generated by revolving the region enclosed by the parabola y^2=4x and the line y=x revolved about the y-axis?

1 Answer
Jul 8, 2017

(128pi)/15

Explanation:

Consider an horizontal strip of thickness delta y revolved around y axis. The volume of the solid so generated would be pi x_1^2 delta y - pi x_2^2 deltay. This is illustrated in the figure below.

The volume of the solid generated by revolving the who region bounded by the given parabola and the straight line would thus be

int_(y=0) ^4 pi x_1^2 delta y - pi x_2^2 deltay. Substituting y= x_1 and y^2 /4 = x_2, te given integral would become

pi int_0^4 (y^2 -y^4 /16 ) dy

pi [ y^3 /3 -y^5 /80]_0^4

pi[64/3 -1024/80] = 64pi(1/3 -1/5)= (128pi)/15