15.1 Double Integrals over Rectangles

Iterated Intergral: $\int_{x=a}^{x=b} \int_{y=c}^{y=d} f(x,y) \, dy \, dx$

• Fubini’s Theorem: (Reversible) $\iint f(x,y)\,dy\,dx = \iint f(x,y)\,dx\,dy$
• Short hand: $\iint_D f(x,y)\,dA$ ($D=[a,b] \times [c,d]$ means $D=\{(x,y) | a \le x \le b, c \le y \le d\}$)
  • Average Value: $f_{avg}=\frac{1}{\text{Area}(R)} \iint_{R} f(x,y) \, dA$
• Area: $\text{Area}(D)=A(D)=\iint_D 1 \, dA$
• Area Estimation: If $m\le f(x,y) \le M$, then $mA(D)\le \iint_D f \, dA \le MA(D)$

15.2 Double Integrals over General Regions

Type I: Plane $D=[a,b] \times [g_1(x), g_2(x)]$ lies between two continuous functions of $x$

• Volume: $\int_a^b\int_{g_1(x)}^{g_2(x)} f(x,y)\, dy \, dx$

Type II: Plane $D=[h_1(x),h_2(x)] \times[c,d]$ lies between two continuous functions of y

• Volume: $\int_c^d\int_{h_1(x)}^{h_2(x)} f(x,y)\, dx \, dy$

15.3 Double Integrals in Polar Coordinates

Polar Rectangle: $R=\{(r, \theta) | a \le r \le b, \alpha \le \theta \le \beta\}$

If $f$ is continuous on polar rectangle $R$ where $0 \le \beta - \alpha \le 2\pi$

• $\iint_{R} f(x,y) \, dA = \int_{\alpha}^{\beta} \int_{a}^{b} r \cdot f(r \cos \theta, r \sin \theta) \, dr \, d\theta$
• Don’t forget to add the $r\cdot$ term!

If $R=\{(r, \theta) \, | \, h_1(\theta) \le r \le h_2(\theta), \, \alpha \le \theta \le \beta\}$

• $\iint_{R} f(x,y) \, dA = \int_{\alpha}^{\beta} \int_{h_1(\theta)}^{h_2(\theta)} r \cdot f(r \cos \theta, r \sin \theta) \, dr \, d\theta$

Intuition: The r and θ define the x-y plane’s shape, and the function is the z-axis height

• Examples