Essential Concepts

Double Integrals over Basic Regions

• We can use a double Riemann sum to approximate the volume of a solid bounded above by a function of two variables over a rectangular region. By taking the limit, this becomes a double integral representing the volume of the solid.
• Properties of double integral are useful to simplify computation and find bounds on their values.
• We can use Fubini’s theorem to write and evaluate a double integral as an iterated integral.
• A general bounded region [latex]D[/latex] on the plane is a region that can be enclosed inside a rectangular region. We can use this idea to define a double integral over a general bounded region.
• To evaluate an iterated integral of a function over a general nonrectangular region, we sketch the region and express it as a Type I or as a Type II region or as a union of several Type I or Type II regions that overlap only on their boundaries.

Applications of Double Integrals

• Double integrals are used to calculate the area of a region, the volume under a surface, and the average value of a function of two variables over a rectangular region.
• We can use double integrals to find volumes, areas, and average values of a function over general regions, similarly to calculations over rectangular regions.
• We can use Fubini’s theorem for improper integrals to evaluate some types of improper integrals.

Double Integrals in Polar Coordinates

• To apply a double integral to a situation with circular symmetry, it is often convenient to use a double integral in polar coordinates. We can apply these double integrals over a polar rectangular region or a general polar region, using an iterated integral similar to those used with rectangular double integrals.
• The area [latex]dA[/latex] in polar coordinates becomes [latex]rdrd\theta[/latex].
• Use [latex]x=r\cos\theta[/latex], [latex]y=r\sin\theta[/latex], and [latex]dA=rdrd\theta[/latex] to convert an integral in rectangular coordinates to an integral in polar coordinates.
• Use [latex]r^{2}=x^{2}+y^{2}[/latex] and [latex]\theta=\tan^{-1}\left(\frac{y}{x}\right)[/latex] to convert an integral in polar coordinates to an integral in rectangular coordinates, if needed.
• To find the volume in polar coordinates bounded above by a surface [latex]z=f(r,\theta)[/latex] over a region on the [latex]xy[/latex]-plane, use a double integral in polar coordinates.

Key Equations

• Double integral
  [latex]\underset{R}{\displaystyle\iint} f(x,y)dA=\underset{m,n\to{\infty}}{\lim}\displaystyle\sum_{i=1}^{m}\displaystyle\sum_{j=1}^{n}f(x_{ij}^{\ast},y_{ij}^{\ast})\Delta{A}[/latex]
• Iterated integral
  [latex]\displaystyle\int_{a}^{b}\displaystyle\int_{c}^{d} f(x,y)dxdy=\displaystyle\int_{a}^{b}\left[\displaystyle\int_{c}^{d}f(x,y)dy\right]dx[/latex]  OR
  [latex]\displaystyle\int_{c}^{d}\displaystyle\int_{b}^{a} f(x,y)dxdy=\displaystyle\int_{c}^{d}\left[\displaystyle\int_{a}^{b}f(x,y)dx\right]dy[/latex]
• Average value of a function of two variables
  [latex]f_{\text{ave}}=\frac{1}{\text{Area R}}\underset{R}{\displaystyle\iint} f(x,y)dxdy[/latex]
• Iterated integral over a Type I region
  [latex]\displaystyle\iint_{D} f(x,y)dA=\displaystyle\iint_{D} f(x,y)dydx=\displaystyle\int_{a}^{b}\left[\displaystyle\int_{g_{1}(x)}^{g_{2}(x)}f(x,y)dy\right] dx[/latex]
• Iterated integral over a Type II region
  [latex]\displaystyle\iint_{D} f(x,y)dA=\displaystyle\iint_{D} f(x,y)dydx=\displaystyle\int_{c}^{d}\left[\displaystyle\int_{h_{1}(y)}^{h_{2}(y)}f(x,y)dx\right] dy[/latex]
• Double integral over a polar rectangular region
  [latex]\underset{R}{\displaystyle\iint} f(r,\theta)dA = \underset{m,n\to\infty}{\lim}\displaystyle\sum_{i=1}^{m}\displaystyle\sum_{j=1}^{n}f({r_{ij}}^{\ast},{\theta_{ij}}^{\ast})\Delta{A} =\underset{m,n\to\infty}{\lim}\displaystyle\sum_{i=1}^{m}\displaystyle\sum_{j=1}^{n}f({r_{ij}}^{\ast},{\theta_{ij}}^{\ast}){r_{ij}}^{\ast}\Delta{r}\Delta{\theta}[/latex]
• Double integral over a general polar region
  [latex]\underset{D}{\displaystyle\iint} f(r,\theta)r dr d\theta=\displaystyle\int_{\theta=\alpha}^{\theta=\beta} \displaystyle\int_{r=h_{1}(\theta)}^{r=h_{2}(\theta)}f(r,\theta)r dr d\theta[/latex]

Glossary

double Riemann Sum

of the function [latex]f(x,y)[/latex] over a rectangular region [latex]R[/latex] is [latex]\displaystyle\sum_{i=1}^{m} {} \displaystyle\sum_{j=1}^{n} {f({x^{*}}_{i,j}, {y^{*}}_{i,j})}[/latex] where [latex]R[/latex] is divided into smaller sub rectangles [latex]R_{ij}[/latex] and [latex]({x^{*}}_{i,j}, {y^{*}}_{i,j})[/latex] is an arbitrary point in [latex]R_{ij}[/latex]

double Integral

of the function [latex]f(x,y)[/latex] over the region [latex]R[/latex] in the [latex]xy[/latex]-plane is defined as the limit of a double Riemann sum, [latex]\underset{R}{\displaystyle\iint} f(x,y)dA=\underset{m,n\to{\infty}}{\lim}\displaystyle\sum_{i=1}^{m}\displaystyle\sum_{j=1}^{n}f(x_{ij}^{\ast},y_{ij}^{\ast})\Delta{A}[/latex]

Fubini’s Theorem

if [latex]f(x,y)[/latex] is a function of two variables that is continuous over a rectangular region [latex]R = \{(x,y)\in{\mathbb{R}}^{2}|a\leq x\leq b,c\leq y\leq d\}[/latex], then the double integral of [latex]f[/latex] over the region equals an iterated integral,

[latex]\underset{R}{\displaystyle\iint} f(x,y)dxdy={\displaystyle\int_{a}^{b}}{\displaystyle\int_{c}^{d} {f(x,y){dx}{dy}}}={\displaystyle\int_{c}^{d}}{\displaystyle\int_{a}^{b} {f(x,y){dx}{dy}}}[/latex]

improper double integral

a double integral over an unbounded region or of an unbounded function

iterated Integral

for a function [latex]f(x,y)[/latex] over the region [latex]\bf{R}[/latex] is

[latex]{\displaystyle\int_{a}^{b}}{\displaystyle\int_{c}^{d} {f(x,y){dx}{dy}}}={\displaystyle\int_{a}^{b}}\left[{\displaystyle\int_{c}^{d} {f(x,y){dy}}}\right]{dx}[/latex]

[latex]{\displaystyle\int_{a}^{b}}{\displaystyle\int_{c}^{d} {f(x,y){dx}{dy}}}={\displaystyle\int_{c}^{d}}\left[{\displaystyle\int_{a}^{b} {f(x,y){dx}}}\right]{dy}[/latex]

polar rectangle

the region enclosed between the circles [latex]r=a[/latex] and [latex]r=b[/latex] and the angles [latex]\theta = \alpha[/latex] and [latex]\theta = \beta[/latex]; it is described as [latex]{\bf{R}}=\{(r,{\theta}) | a{\leq}r{\leq}b, {\alpha}{\leq}{\theta}{\leq}{\beta}\}[/latex]

Type I

a region [latex]\bf{D}[/latex] in the [latex]xy[/latex]-plane is Type I if it lies between two vertical lines and the graphs of two continuous functions [latex]g_{1}(x)[/latex] and [latex]g_{2}(x)[/latex]

Type II

a region [latex]\bf{D}[/latex] in the [latex]xy[/latex]-plane is Type II if it lies between two horizontal lines and the graphs of two continuous functions [latex]h_{1}(y)[/latex] and [latex]h_{2}(y)[/latex]