4.5 Applications of Double Integrals

Applications of Double Integrals

Introduction

Area

Mass

Probability

Center of Mass

Suppose that we have a flat, thin object (called a lamina) whose density varies across the object. We can think of the density on a lamina as a measure of mass per unit area. As an example, consider a circular plate $D$ of radius 1 cm centered at the origin whose density $\delta$ varies depending on the distance from its center so that the density in grams per square centimeter at point $(x, y)$ is

Partition the plate into subrectangles $R_{ij}$, where $1 \leq i \leq m$ and $1 \leq j \leq n$, so that each subrectangle has area $\Delta A$. Select a point $(x_{ij}^*,y_{ij}^*)$ in $R_{ij}$ for each $i$ and $j$. Write a couple of sentences to explain the meaning of the quantity $\delta(x_{ij}^*,y_{ij}^*) \Delta A$.

The volume of a solid with constant height is given by the area of the base times the height. Hence, we may interpret the area of the region $D$ as the volume of a solid with base $D$ and of uniform height 1. That is, the area of $D$ is given by $\iint_D 1 \, dA$. Write an iterated integral whose value is $\iint_D 1 \, dA$.

Evaluate the iterated integral from (a). What does the result tell you?

Theoretically, the components might last forever, so the domain $D$ of the function $f$ is the set $D$ of all $(x,y)$ such that $x \ge 0$ and $y \ge 0$. To show that $f$ is a probability density function on $D$ we need to demonstrate that

, or that

. Use your knowledge of improper integrals to verify that $f$ is indeed a probability density function.

Assume that the smoke detector fails only if both of the supplied components fail. To determine the probability that a randomly selected detector will fail within one year, we will need to determine the probability that the life span of each component is between 0 and 1 years. Set up an appropriate iterated integral, and evaluate the integral to determine the probability.

What is the probability that a randomly chosen smoke detector will fail between years 3 and 7?

Suppose that the manufacturer determines that one of the components is more likely to fail than the other, and hence conjectures that the probability density function is instead $f(x,y) = K e^{-x} e^{-2y}.$ What is the value of $K$?

Set up an iterated integral that computes the total mass of the lamina.

Assume the total mass of the lamina is 14. Set up two iterated integrals that represent the coordinates of the center of mass of the lamina.