Question

Write the integral in one variable to find the volume of the solid obtained by rotating the first-quadrant region bounded by y = 0.5x² and y = x about the line x = 5.

Answer 1

Shell method :

we know that volume of the solid generated by revolving the region bounded by y = f(x) and y = g(x) between x = a and x = b

about the line x = x is given by,

--------------------------------------------------1)

where f(x) is the top curve above g(x) between x = a and x = b

and a < b < c

we have y = 0.5x² and y = x hence we can write,

As given region is bounded in first quadrant

we have x = 0 and x = 2 which is in first quadrant hence we can say that region is bounded between x = 0 and x = 2

we can graph y = 0.5x² and y = x between x = 0 and x = 2 as below:

we can see that y = x is the top curve above y = 0.5x² hence f(x) = x and g(x) = 0.5x²

As region is bounded between x = 0 and x = 2 we have a = 0 and b = 2

we have to rotate the region about x = 5 hence c = 5

as 0 < 2 < 5 we can say that a < b < c

Hence using formula 1) we can say that volume is given by,

we can evaluate the integral as below: