Double Integrals:

Evaluate the Iterated Integrals:

• $\int_1^4 \int_{0}^2 (6x^2y-2x) dy dx$
• $\int_{\pi/6}^{\pi/4} \int_{-1}^{5} \cos(s) dt ds$
• $\int_1^3 \int_1^{5} \frac{\ln y}{xy} dy dx$
• $\int_0^1 \int_0^1 \sqrt{x+y} dy dx$

Double Integrals:

Evaluate the Double Integrals:

• $\iint_D \sin(x-y) dA$, $D = \{(x,y): 0 \leq x \leq \pi/2, 0 \leq y \leq \pi/2 \}$
• $\iint_D y e^{-xy} dA$, $D = \{(x,y): 0 \leq x \leq 2, 0 \leq y \leq 3 \}$

Volume

Find the volume of the solid that lies under the hyperbolic paraboloid $z = 3y^2 - x^2 + 2$ and above the rectangle $R = \{(x,y): -1 \leq x \leq 1, 1 \leq y \leq 2 \}$.

Find the volume of the solid enclosed by the surface $z = 1 + e^x \sin y$ and the planes $x = \pm 1, y = 0, y = \pi,$ and $z = 0$.

Symmetry

Use symmetry to evaluate the double integrals:

• $\iint_R \frac{xy}{1+x^4} dA, R = \{ (x,y): -1 \leq x \leq 1, 0 \leq y \leq 1 \}$
• $\iint_R (1+ x^2 \sin y + y^2 \sin x) dA, R = [-\pi, \pi] \times [-\pi, \pi]$