Find the volume under the surface

Question

Find the volume under the surface $z=f(x,y)$ over the rectangle R, where $f(x,y)=x^4+xy+y^3$ and $R=[1,2]\times[0,2]$

Answer 1

With your comment, you're most of the way along to a solution. You correctly realize that this problem is to find $$ \int\int f(x, y)\,dx\,dy $$ over the domain defined by (I presume) $1\le x\le 2, 0\le y\le 2$. With this in mind, what bounds do you have for the innermost integral (the $x$ part) and what bounds do you have for the outer integral (the $y$ part)? In other words, what will $a, b, c, d$ be in $$ \int_a^b\int_c^d f(x, y)\,dx\,dy $$ Once you've set these values, the integral should be easy. The inner one will evaluate to sone function involving just $y$ and integrating that with respect to $y$ will give you the number you want.