# Double Integrals

### Instructions

- This workshop is a brief review or introduction to double integrals.
- You can attempt to solve the problems first, and then check your answers. Each example also has a video explaining the problem.
- If you want to see more videos on this topic you can see the material in Sections 15.1–15.9 of our Math 251 course.

## Two Dimensional Integrals Over Rectangles

### Examples

**Directions:**You should attempt to solve the problems first, and then you can check your solution and watch the video for an explanation.

- Compute \(\displaystyle \int_{y=2}^{3}\int_{x=1}^{2}x^2y\,dxdy.\)
- Compute \(\displaystyle \int_{x=1}^{2}\int_{y=2}^{3}x^2y\,dydx\)

- Assume a plate lies on the rectangle \([1,2]\times[2,3]\) and has weight density \(yx^2 \frac{\textnormal{pounds}}{\textnormal{foot}^2}\) What is the weight of this plate?
- What is the weight of the portion of this plate that lies on \([1,1.5]\times[2.5,3]?\)

- Let \(f(x,y) = 6x^2 y\) on \([0,1]\times[0,1]\) and \(0\) everywhere else, be the p.d.f. of the two random variables \(X\) and \(Y.\) Verify that \(f\) is a p.d.f.
- What is the probability of the event \(0<X<\dfrac{3}{4}\) and \(\dfrac{1}{3}<Y<1?\)

## Two Dimensional Integrals Over Other Regions

### Examples

**Directions:**You should attempt to solve the problems first, and then you can check your solution and watch the video for an explanation.

- Compute \(\displaystyle \int_{x=0}^{1}\int_{y=x^2}^{x}x^2 y\, dydx.\)
- Compute \(\displaystyle \int_{y=0}^{1}\int_{x=y}^{\sqrt{y}}x^2y\,dxdy.\)

- Let \(f(x,y) = e^{-x-y}\) on \([0,\infty)\times[0,\infty).\) What is the integral of \(f\) over the region \(x+y\leq 1?\)
- Let \(f(x,y) = e^{-x-y}\) on \([0,\infty)\times[0,\infty).\) What is the integral of \(f\) over the region \(x+y\leq z\), where \(z\) is any non–negative number?

## Changing Order of Integration

### Examples

**Directions:**You should attempt to solve the problems first, and then you can check your solution and watch the video for an explanation.

- What is Fubini's Theorem?
- Compute \(\displaystyle \int_{x=0}^{1}\int_{y=x}^{1}e^{y^2}\,dydx\)

- Find \(\displaystyle \int_{x=1/2}^{2}\int_{y=1/x}^{2}y\cos(xy)\,dydx.\)

## Two Dimensional Integrals – Change of Variables

### Examples

**Directions:**You should attempt to solve the problems first, and then you can check your solution and watch the video for an explanation.

- Find the integral of \(f(x,y) = x^2 + y^2\) over the disk of radius \(1\) centered at the origin.

- What is the change of variables formula in two variables?