Units

Unit 6: Double Integrals Over the General Region

The Concept

Now, you should engage with the 3D plot below to understand the double integral over the general region (i.e., non-rectangular region). There are two types of double integrals.

  • Type I double integral: [latex]\int_{a}^{b} \int_{h(x)}^{g(x)} f(x,y)\,dy dx[/latex], where [latex]x=a[/latex] and [latex]x=b[/latex] are the lower and upper bounds of [latex]x[/latex]; [latex]y=h(x)[/latex] and [latex]y=g(x)[/latex] are the lower and upper bounds of [latex]y[/latex].
  • Type II double integral: [latex]\int_{a}^{b} \int_{h(x)}^{g(x)} f(x,y)\,dx dy[/latex], where [latex]y=a[/latex] and [latex]y=b[/latex] are the lower and upper bounds of [latex]y[/latex]; [latex]x=h(y)[/latex] and [latex]x=g(y)[/latex]are the lower and upper bounds of [latex]y[/latex].

You may notice that the bounds of outer integrals ([latex]a[/latex] and [latex]b[/latex]) for both Type I and Type II integrals are constants; these two integrals are “symmetric” – if you switch [latex]x[/latex] and [latex]y[/latex] in Type I, you get Type II and vice versa.

The Plot

Now, you should engage with the plot below to understand double integrals with general regions[1]. Follow the steps below to apply changes to the plot and observe the effects:

  1. Assume you have a Type I integral [latex]\int_{0}^{1} \int_{-x}^{x^2} y^2 x\,dy dx[/latex]. Input [latex]y^2 x[/latex] into the [latex]f(x,y)[/latex] input function section.
  2. Input [latex]\textrm{If} (0 \leq x \leq 1, x^2)[/latex] into the Upper [latex]y[/latex] function, i.e., [latex]g(x)[/latex] section.
  3. Input [latex]\textrm{If} (0 \leq x \leq 1, -x)[/latex] into the lower [latex]y[/latex] function, i.e., [latex]h(x)[/latex] section.
  4. Use the slider for the value of x to see the change of the area of the cross-section, [latex]A(x)[/latex].
  5. The result of this double integral is dynamically calculated at the bottom.
  6. You can also use this plot for the Type II integral by switching [latex]x[/latex] and [latex]y.[/latex]

Self-Checking Questions

Check your understanding by solving the following questions[2]:

  1. [latex]\int_{0}^{1} \int_{2\sqrt{x}}^{2\sqrt{x}+1} xy+1\,dy dx[/latex].
  2. [latex]\int_{0}^{1} \int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}} 2x+4x^3\,dx dy[/latex]

Use the graph to find the answers to these questions.


  1. Made with GeoGebra, licensed Creative Commons CC BY-NC-SA 4.0.
  2. Gilbert Strang, Edwin “Jed” Herman,  OpenStax, Calculus Volume 3,  Calculus Volumes 1, 2, and 3 are licensed under an Attribution-NonCommercial-Sharealike 4.0 International License (CC BY-NC-SA).

License

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3D Interactive Plots for Multivariate Calculus by Dr. Na Yu, Ryerson University is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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