Calculating the volume of a solid with a double integral

Question

The solid lies under the hyperboloid $z = xy$ and above the triangle in the $xy-$plane with vertices $(1, 2),(1, 4),$ and $(5, 2)$. Find the volume of the given solid.

This is the work I have done so far:

$\int\limits_1^5\int\limits_1^b f(x,y) dydx$

where $b=-0.5x+4.5$

and $f(x,y)=xy$

This gives me the answer of $42$, while the correct answer is $24$. I believe my error is in the bounds of the integral, but I can't figure out where it is. Can someone tell me what the error is with my integral?

Answer 1

You first obtain $\int_{2}^{4.5-0.5x}ydy=\frac{16.25-4.5x+0.25x^2-4}{2}$

Then $\int_{1}^{5}\frac{16.25x-4.5x^2+0.25x^3-4x}{2}dx=24.$

Answer 2

The error is in the integral being evaluated over y.

$\int\limits_1^5 \int\limits_2^b f(x,y) \, dydx$