Originally Posted by

**hollywood** I think you have the region right - its boundaries are the vertical lines $\displaystyle x=1$ and $\displaystyle x=3$ on the right and left, the horizontal line $\displaystyle y=0$ on the bottom, and the curve $\displaystyle y=x^3$ on the top.

The way I learned it was to divide the region into vertical rectangles. These rectangles have width $\displaystyle dx$ and height $\displaystyle x^3$, so the sum of $\displaystyle x^3\,dx$ is the area of the region. In the limit, it's the integral of $\displaystyle x^3\,dx$ and the limits are $\displaystyle x=1$ on the left and $\displaystyle x=3$ on the right. So the area is given by:

$\displaystyle \int_1^3x^3\,dx$.

Hopefully that answers your question.

- Hollywood