1. ## Double integrals

I've just started learning double integration and I'm having trouble doing the following questions.

Use double integration to find the area of the following regions.

1) The region bounded by $y=x^3$ and $y=x^2$

Would this be how I'd start it off?

$\int_0^1\int_{x^2}^{x^3} \ dy \ dx$

2) The region bounded by $y=\sqrt{x}, y=x$ and $y=\frac{x}{2}$

$\int_0^1\int_{x/2}^x \ dy \ dx + \int_1^2\int_{x/2}^{\sqrt{x}} \ dy \ dx$

2. You really should draw the regions, it helps you to decide on the region of integration.

For the first one be careful, in the region $0 \leq x \leq 1$ it's actually $x^3 \leq y \leq x^2$.

So your double integral should be

$\displaystyle \int_0^1{\int_{x^3}^{x^2}{\,dy}\,dx}$.

For the second, the region actually goes from $0 \leq x \leq 4$.

So your double integral should be

$\displaystyle \int_0^1{\int_{\frac{x}{2}}^{x}{\,dy}\,dx} + \int_1^4{\int_{\frac{x}{2}}^{\sqrt{x}}{\,dy}\,dx}$.

3. Is the second integral okay?

4. No, see my edit.