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. 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:

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). 