# Region of integration for a double integral

My book gives the integral $$\int_0^2 \int_x^2 x\sqrt{1+y^3} dydx$$

And the directions say to sketch the region of integration, and evaluate the integral by switching the order of integration.

Once I have the region sketched out I think I know how to find the area by changing it to $dxdy$, but I'm lost at how to draw the region. Up to now, when we had iterated integrals representing regions, they were always of the form $$\int_b^a \int_{g(x)}^{f(x)} dydx$$or $$\int_b^a \int_{g(y)}^{f(y)} dxdy$$

I know how to draw the region for these where the integrand is $1$, but I've never encountered one where the integrand isn't $1$.

• The region doesn't depend on the integrand! – Hans Lundmark Nov 17 '15 at 19:24
• It doesn't matter, because the region of integration only represents the domain of the function, not the function itself – Dylan Nov 17 '15 at 19:29
• @Dylan So I just pretend like the integrand doesnt exist and So the region is just bounded vertically by y = 2 and y = x, and horizontally by x = 2 and x = 0? ? – Ovi Nov 17 '15 at 19:34
• Yes. That region is the same no matter the function. Unless the function is undefined there, but that's another issue. – Dylan Nov 17 '15 at 19:52