Double Integrals

Double integrals over rectangular regions

The function $f(x,y)$ is defined on a rectangular region $R$ $$R:~a\leq x\leq b,~c\leq y\leq d.$$

We divide the region into small grids, and for each grid, we pick a point $(x_k,y_k)$ and use $f(x_k,y_k)\Delta A_k$ to approximate the intergal over this small rectangular. Let $$S_n=\sum_{k=1}^nf(x_k,y_k)\Delta A_k=\sum_{k=1}^nf(x_k,y_k)\Delta x_k \Delta y_k.$$ If $\lim\limits_{n\rightarrow\infty}S_n$ does not change no matter what choices are made, we say $f$ is integral, and the limit is called the double integral of $f$ over $R$, written as $$\iint_Rf(x,y)dA,\qquad\mbox{or }\qquad \iint_Rf(x,y)dxdy.$$

Which function is integrable

▶ A continuous function is integrable;

▶ A function that is discontinuous only on a finite number of points or smooth curves is also integrable.

Double integrals as volumes

Fubini’s theorem for calculating double integrals

Calculate the volume under the plane $z = 4 - x - y$ over the rectangular region $R: 0 \leq x \leq 2,~0 \leq y \leq 1$ in the $xy$-plane.

Fubini's Theorem

Theorem (Theorem 1)

If $f(x, y)$ is continuous throughout the region $R: a \leq x \leq b, c \leq y \leq d$, then $$\iint_Rf(x,y)dA=\int_c^d\int_a^bf(x,y)dxdy=\int_a^b\int_c^df(x,y)dydx.$$

Example: $$\int_0^2\int_{-1}^1(100-6x^2y)dydx=\int_{-1}^1\int_0^2(100-6x^2y)dxdy$$

Example:

Find the volume of the region bounded above by the elliptical paraboloid $z = 10 + x^2 + 3y^2$ and below by the rectangle $R: 0\leq x \leq 1, 0\leq y \leq 2$.

Solution:

Double integrals over bounded, nonrectangular regions

Double Integrals as Volumes

Fubini’s theorem (stronger Form)

Let $f(x, y)$ be continuous over a region $R$.

▶ If $R$ is defined by $a\leq x\leq b$, $g_1(x)\leq y\leq g_2(x)$ with $g_1$ and $g_2$ continuous on $[a,b]$, then $$\iint_Rf(x,y)dA=\int_a^b\int_{g_1(x)}^{g_2(x)}f(x,y)dydx.$$

▶ If $R$ is defined by $c\leq y\leq d$, $h_1(y)\leq x\leq h_2(y)$ with $h_1$ and $h_2$ continuous on $[c,d]$, then $$\iint_Rf(x,y)dA=\int_c^d\int_{h_1(y)}^{h_2(y)}f(x,y)dxdy.$$

Example: Find the volume of the prism whose base is in the $xy$-plane bounded by $y=0$, $y = x$, and $x = 1$ and whose top lies in the plane $z=3-x-y$.

Example

Calculate

where $R$ is bounded by $y=0$, $y=x$, and $x=1$.

Find limits of integration

Using vertical cross-sections

▶ Sketch the region of integration

▶ Find the y-limits of integration

▶ Find the x-limits of integration

Example: Sketch the region of integration for the integral

and write an equivalent integral with the order of integration reversed.

Solution: $$\int_0^2\int_{x^2}^{2x}(4x+2)dydx=\int_0^4\int_{y/2}^{\sqrt y}(4x+2)dxdy$$

Properties of double integrals

If $ƒ(x, y)$ and $g(x, y)$ are continuous on the bounded region $R$, then the following properties hold.

1. Constant Multiple: $\iint_Rcf(x,y)dA=c\iint_Rf(x,y)dA$ (any number $c$)

2. Sum and Difference: $\iint_R(f(x,y) \pm g(x,y))dA=\iint_Rf(x,y)dA \pm \iint_Rg(x,y)dA$

3. Domination:

   (a) $\iint_Rf(x,y)dA \geq 0$ if $f(x,y) \geq 0$ on $R$

   (b) $\iint_Rf(x,y)dA \geq \iint_Rg(x,y)dA$ if $f(x,y) \geq g(x,y)$ on $R$

4. Additivity: $\iint_Rf(x,y)dA=\iint_{R_1}f(x,y)dA+\iint_{R_2}f(x,y)dA$ if $R$ is the union of two nonoverlapping regions $R_1$ and $R_2$.

Example:

Find the volume of the wedgelike solid that lies beneath $z=16-x^2-y^2$ and above the region $R$ bounded by the curve $y = 2\sqrt{x}$, the line $y = 4x - 2$, and the $x$-axis.

Solution: $$\text{Volume}=\int_0^2\int_{y^2/4}^{(y+2)/4}(16-x^2-y^2)dxdy={20803\over1680}$$

Area by double integration ($f(x,y)=1$)

definition The area of a closed bounded plane region $R$ is $$A=\iint_RdA$$

Example: Find the area of the region $R$ bounded by the parabola $y=x^2$ and the line $y=x+2$

solution: $$\text{Area}=\iint_RdA=\int_{-1}^{2}\int_{x^2}^{x+2}dydx={9\over2}$$

Example: Find the area of the playing field described by $$R:~-2\leq x\leq 2,~-1-\sqrt{4-x^2}\leq y\leq 1+\sqrt{4-x^2},$$ using (a) Fubini's theorem, (b) simple geometry.

Solution:

(a) $\text{Area}=\int_{-2}^{2}\int_{-1-\sqrt{4-x^2}}^{1+\sqrt{4-x^2}}dydx=4\pi+8$

(b) $\text{Area}=Area(two\ semicircles)+Area(rectangle)=2^2\times \pi+4\times 2=4\pi+8$

Average Value

Example: Find the average value of $f(x,y)=x\cos xy$ over the rectangle $R:0\leq x\le \pi,~0\leq y\leq 1$.

Solution: $\iint_RfdA=\int_0^\pi\int_0^1x\cos(xy)dydx=2$

Double integrals in the Polar form

Finding limits of integration

▶ Sketch the region of integration

▶ Find the r-limits of integration

▶ Find the θ-limits of integration

Example: Find the limits of integration for integrating $f(r, u)$ over the region $R$ that lies inside the cardioid $r = 1 + \cos \theta$ and outside the circle $r = 1$. $$\iint_Rf(r,\theta)dA=\int_{-\pi/2}^{\pi/2}\int_{1}^{1+cos\theta}f(r,\theta)rdrd\theta$$

Area in polar coordinates ($f=1$)

Example: Find the area enclosed by the lemniscate $r^2=4\cos2\theta$. Solution: $$\iint_R rdrd\theta=4\times \int_0^{\pi/4}\int_{0}^{2\sqrt{cos2\theta}}rdrd\theta=4$$

Changing cartesian integral to polar integral

Example: Evaluate $$\iint_Re^{x^2+y^2}dydx$$ where $R$ is the semicircular region bounded by the $x$-axis and the curve $y=\sqrt{1-x^2}$.

Solution: $$\iint_Re^{x^2+y^2}dydx=\int_0^{\pi}\int_{0}^{1}e^{r^2}rdrd\theta={(e-1)\pi\over2}$$

Example: Evaluate the integral $$\int_0^1\int_0^{\sqrt{1-x^2}}(x^2+y^2)dydx.$$

Solution: $$\iint_R(x^2+y^2)dydx=\int_0^{\pi/2}\int_{0}^{1}(r^2)rdrd\theta={\pi/8}$$

Example3:

Find the volume of the solid region bounded above by the paraboloid $z = 9 - x^2 - y^2$ and below by the unit circle in the $xy$-plane

Solution: $$\iint_R(9-x^2-y^2)dydx=\int_0^{2\pi}\int_{0}^{1}(9-r^2)rdrd\theta={17\pi/2}$$