Use a double integral to find the volume of the solid bounded by graphs of the equations given by: z=xy^2, where: z>0, x>0, 5x<y<2.

Gorlandint

Gorlandint

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2022-08-16

Multivariable calculus double integration volume question
Use a double integral to find the volume of the solid bounded by graphs of the equations given by:
z = x y 2 ,  where:  z > 0 x > 0 5 x < y < 2
My problem is finding the limits of integration. I know my f ( x , y )... my guess is my y integration is going from 5x going to 2. But how do I find the x limitations?

Answer & Explanation

Skylar Beard

Skylar Beard

Beginner2022-08-17Added 11 answers

Step 1
0 < 5 x < y < 2 , 0 < z < x y 2 x ( 0 , 2 5 ) , y ( 5 x , 2 ) , z ( 0 , x y 2 )
Step 2
The lower bound on x is 0, the upper bound is y/5, which in turn has an upper bound of 2/5.
0 < 5 x < y < 2 , 0 < z < x y 2 d z d y d x = 0 2 / 5 5 x 2 x y 2 d y d x = 0 2 0 y / 5 x y 2 d x d y

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