Volume under a Surface

Calculus Level 4

Calculate the volume of the solid below f ( x , y ) = 4 e x y 2 f(x,y)=4{ e }^{ x }{ y }^{ 2 } and above the region R R in the x y xy -plane bound by y = 5 y=5 , x = ln ⁡ 3 x=\ln { 3 } , and x = ln ⁡ y x=\ln { y } .


The answer is 152.

Liam P by Liam P , 22, Canada

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1 solution

First Last
Nov 10, 2017

The region of integration is:

The integral of the function f ( x , y ) = 4 e x y 2 f(x,y)=4e^xy^2 on this region is 4 ∫ 3 5 ∫ ln ⁡ 3 ln ⁡ y e x y 2 d x d y = 4 ∫ 3 5 y 3 − 3 y 2 d y = 152 \displaystyle 4 \int_3^5 \int_{\ln{3}}^{\ln{y}}e^xy^2dxdy = 4\int_3^5y^3-3y^2dy = \boxed{152}

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How to find bounds for y?

D S - 3 years, 6 months ago

@Jasper Braun

D S - 3 years, 6 months ago

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Looking at that region I kept the x varying from ln(3) to ln(y) with y going from 3 to 5. Imagine that you are picking all y between 3 and 5 and using that for bounds of the inner integral (the one with respect to x). Look into multivariable calc for a better explanation on bounds of double integrals.

First Last - 3 years, 6 months ago

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