Repeated integrals

Exercise

1.

Evaluate the following double integrals．

(a)

$${\iint_{\Omega}x^2 dxdy, \ \Omega: -1 \leq x \leq 1, \ 0 \leq y \leq 3}$$

(b)

$${\iint_{\Omega}e^{x+y} dxdy, \ \Omega: 0 \leq x \leq 1, \ 0 \leq y \leq x}$$

(c)

$${\iint_{\Omega}\sqrt{xy} dxdy, \ \Omega: 0 \leq y \leq 1, \ y^2 \leq x \leq y}$$

(d)

$${\iint_{\Omega}(4 - y^2) dxdy, \ \Omega}$$ is the region bounded by $y^2 = 2x$ and $y^2 = 8 - 2x$ .

2.

Interchange the order of integration.．

(a)

$${\int_{0}^{1}\int_{x^4}^{x^2}f(x,y)dydx}$$

(b)

$${\int_{0}^{1}\int_{-y}^{y}f(x,y)dxdy}$$

(c)

$${\int_{1}^{4}\int_{x}^{2x}f(x,y)dydx}$$

3.

Evaluate the following double integrals．

(a)

$${\int_{0}^{1}\int_{y}^{1}e^{y/x}dxdy}$$

(b)

$${\int_{0}^{1}\int_{x}^{1}e^{y^2}dydx}$$

(c)

$${\int_{0}^{1} dy \int_{y}^{\sqrt{y}}\frac{\sin{x}}{x}dx}$$