Applications of Double Integrals¶
In these notes we'll look at a couple examples of applications of double integrals. For the interested students, there are more applications in this section of the book.
Moments and Centers of Mass¶
A familiar problem from Calculus II is to find the center of mass of a lamina: a thin sheet with density. The following video describes how double integrals can be used to help solve this problem.
Video Lecture
1. Example¶
Find the center of mass of a triangular region with vertices $(0,0)$, $(1,0)$, and $(0,2)$, and with density function $\rho(x,y) = 1 + 3x + y$.
Check your answer
$$\left(\frac{3}{8},\frac{11}{16}\right)$$
Video Solution
2. Example¶
Consider the region $D_a = \big\{(x,y) \mid x^2 + y^2 = a^2, y \geq 0 \big\}$, where $a > 0$ is a constant. Suppose this region has density function $\rho(x,y) = K\sqrt{x^2 + y^2}$ for some positive constant $K$. Find the center of mass of this laminar region.
Check your answer
$$\left( 0, \frac{3a}{2\pi} \right)$$
Video Solution
Probability¶
A probability density function on a domain $D \subseteq \mathbb{R}^2$ is a function $f$ that is continuous and non-negative on $D$, and for which
$$
\int\!\!\!\int_D f(x,y)\, dA = 1.
$$
For any sub-region $S \subset D$, the probability that a point $Q(x_0,y_0) \in D$ lies in $S$ is given by
$$
P\big(Q \in S\big) = \int\!\!\!\int_S f(x,y)\, dA.
$$
Video Lecture
3. Example¶
Show that the function $f(x,y) = \frac{1}{\pi}e^{-x^2-y^2}$ is a probability density function on $\mathbb{R}^2$.
Hint
Switch to polar coordinates.
Need more?
To fill out the entire plane, $\theta$ varies from $0$ to $2\pi$, and $r$ varies from $0$ to $\infty$. The integral is separable in polar coordinates, and fairly easy to compute by hand. But beware, the $r$ integral is improper.
Video Solution
Ancillary Question: Having just worked through this exercise, can you deduce the value of this infamous integral?
$$
\int_{-\infty}^\infty e^{-x^2}\, dx
$$
Hint
$$\int\!\!\!\int_{\mathbb{R}^2} e^{-x^2-y^2}\, dA = \int_{-\infty}^\infty\int_{-\infty}^\infty e^{-x^2}e^{-y^2}\,dx\,dy = \left[ \int_{-\infty}^\infty e^{-x^2} \, dx \right] ^2. $$
4. Expected Value¶
Suppose $f$ is a probability density function on a set $D \subseteq \mathbb{R}^2$.
The expected value of a random variable $(X,Y)$ on $D$ is given by $(\mu_x,\mu_y)$, where
$$
\mu_x = \int\!\!\!\int_D xf(x,y)\, dA \ \ \text{and}\ \ \mu_y = \int\!\!\!\int_D yf(x,y)\, dA.
$$
Consider the function
$$
f(x,y) = 4xy
$$
on the domain $D = [0,1] \times [0,1]$. Show that $f$ is a probability density function on $D$. Then compute the expected value.
Check your answer
The expected value is $$\left(\frac{2}{3},\frac{2}{3}\right).$$
Video Solution
Discussion¶
Questions? You can ask them here.