Question

Sketch the regions enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width.
y = sqrt of x
y = (1/3)x
x=16

Answer 1

It doesn't matter, you can do it either way. Either way you end up splitting it up into 2 integrals.

If you integrate with respect to x you would do:
\[\int\limits_{0}^{9}(\sqrt(x)-x/3)dx+\int\limits_{9}^{16}(x/3-\sqrt(x))dx\]

To find the place where you would split the integral up you need to set x/3 equal to sqrtx. Once you solve for x, the answers end up being 0 and 9. Pretty obvious how to split it up at that point.

Answer 2

thank you guys both soo much :)