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

You may notice that the bounds of outer integrals ($a$ and $b$) for both Type I and Type II integrals are constants; these two integrals are “symmetric” – if you switch $x$ and $y$ 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 $\int_{0}^{1} \int_{-x}^{x^2} y^2 x\,dy dx$. Input $y^2 x$ into the $f(x,y)$ input function section.
2. Input $\textrm{If} (0 \leq x \leq 1, x^2)$ into the Upper $y$ function, i.e., $g(x)$ section.
3. Input $\textrm{If} (0 \leq x \leq 1, -x)$ into the lower $y$ function, i.e., $h(x)$ section.
4. Use the slider for the value of x to see the change of the area of the cross-section, $A(x)$.
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 $x$ and $y.$

Self-Checking Questions

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

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

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).