4.3 Iterated Integrals

Iterated Integrals

Introduction

Iterated Integrals

In this activity we will explore the double integral of $f(x,y) = 25-x^2-y^2$ on the rectangular domain $R$ containing the points that satisfy $-3\leq x\leq 3$ and $-4\leq y\leq 4$. As with partial derivatives, we may treat one of the variables in $f$ as constant and think of the resulting function as a function of a single variable. We will now investigate what this means geometrically and algebraically when we integrate with one input variable fixed.

Let $a$ be a fixed value in the interval $[-3,3]$. Compute

.

Viewing $x$ as a fixed constant, use the Fundamental Theorem of Calculus to evaluate the integral

Note that you will be integrating with respect to $y$, and holding $x$ constant. Your result should be a function of $x$ only.

Next, use your result from (a) along with the Fundamental Theorem of Calculus to determine the value of $\int_{-3}^3 A(x) \, dx$.

What is the value of $\displaystyle{\iint_R f(x,y) \, dA}$? Write a sentence to interpret the meaning of this value for two out of the three different ways mentioned in Proposition 4.2.1.

Evaluate $\displaystyle{\iint_R f(x,y) \, dA}$ using an iterated integral. Choose an order for integration by deciding whether you want to integrate first with respect to $x$ or $y$.

Evaluate $\displaystyle{\iint_R f(x,y) \, dA}$ using the iterated integral whose order of integration is the opposite of the order you chose in (a).