Essential Concepts
- 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.
- 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.
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]
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]
- 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]
