Use a double integral to find the volume of the solid bounded by graphs of the equations given by: $$\begin{align}z=xy^2, \text{ where: } &z>0\\&x>0\\&5x<y<2\end{align}$$

My problem is finding the limits of integration. I know my $f(x,y)$ is $xy^2$...my guess is my $y$ integration is going from $5x$ going to $2$. But how do I find the $x$ limitations?


1 Answer 1


$0< 5x < y < 2, 0<z<xy^2 \iff x\in (0, \frac{2}{5}), y\in (5x, 2), z\in(0,xy^2)$

The lower bound on $x$ is $0$, the upper bound is $y/5$, which in turn has an upper bound of $2/5$.

$$\begin{align}\therefore \iiint_{0<5x<y<2, 0<z<xy^2} \operatorname{d}z \operatorname{d} y \operatorname{d} x & = \int_0^{2/5} \!\! \int_{5x}^2 x y^2\operatorname{d} y \operatorname{d} x \\ & = \int_0^2\int_0^{y/5} xy^2 \operatorname{d}x\operatorname{d}y \end{align}$$


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