Double Integrals over General Regions

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Summary and an Important Example

Double Integrals over General Regions

Type I and Type II regions
Examples 1-4
Examples 5-7
Order of Integration

Examples 1-4

In the following video, we will compute Example 1 and Example 2:

Example 1: Evaluate

$\iint_R f(x,y)\,dA,$ where

$f(x,y)= x^2y$ and

$R$ is the upper half of the unit disk (shown here).

Example 2: Evaluate

$\iint_R f(x,y)\,dA,$ where

$f(x,y) = 4xe^{2y}$ and the region

$R$ is bounded by the

$x$ -axis, the

$y$ -axis, the line

$y=2$ and the curve

$y=\ln(x)$ (shown here).

Now, we look at some additional examples.

Example 3: Evaluate the integral

$I \iint _D\, (x+y)\, dA$ where

$D$ consists of all points

$(x,\,y)$ such that

$0 \ \le \ y \ \le \sqrt{9-x^2}\,, \quad 0\ \le \ x \ \le 3\,.$

DO: Graph the region

$D$ before reading further. Then try to set up a Type I iterated integral.

Solution 3: Since

$y^2 \ = \ 9 - x^2$ is a circle of radius

$3$ centered at the origin,

$D$ consists of all points in the first quadrant inside this circle as shown here. This is described as a Type I region, so we

fix $x$ in and integrate with respect to $y$ along the black vertical line as shown, and
then integrate with respect to $x$ from $x=0$ to $x=3$ (all possible black vertical lines).

So the double integral

$I$ becomes the interated integral

$\begin{eqnarray} \int_0^3\left(\int_0^{\sqrt{9-x^2}}\, (x+y)\, dy\right)\,dx &=& \int_0^3\, \Bigl[\, xy +\frac{1}{2}y^2 \,\Bigl]_0^{\sqrt{9 - x^2}}\,dx\\ &=&\int_0^3\, \Bigl( x\sqrt{9 - x^2} +\frac{1}{2} (9 - x^2)\Bigr)\, dx\\ \end{eqnarray}$

DO: Evaluate the above integral. The answer is 18.

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Example 4: Evaluate the integral

$\displaystyle I=\iint_R\, (x+y)\, dA,$ where

$R$ consists of all points

$(x,y)$ with

$0\le y\le 3$ and

$0\le x\le\sqrt{9-y^2}$ .

DO: Graph this region

$R$ . Is it the same as

$D$ above? Set up and evaluate the integral above as a Type II integral over

$R$ , which will have black horizontal lines. You should get the same answer as in Example 3.

The Fundamental Theorem of Calculus

The Indefinite Integral and the Net Change

Substitution

Area Between Curves

Volumes

Integration by Parts

Integrals of Trig Functions

Trig Substitutions

Partial Fractions

Strategies of Integration

Improper Integrals

Differential Equations

Models of Growth

Infinite Sequences

Infinite Series

Integral Test

Comparison Tests

Convergence of Series with Negative Terms

The Ratio and Root Tests

Strategies for testing Series

Power Series

Representing Functions as Power Series

Taylor and Maclaurin Series

Applications of Taylor Polynomials

Partial Derivatives

Multiple Integrals

Iterated Integrals over Rectangles

Double Integrals over General Regions

Examples 1-4