Activity 12.3.2

Let $f(x,y) = 25-x^2-y^2$ on the rectangular domain $R$ with $-3\leq x\leq 3$ and $-4\leq y\leq 4$.

1. Viewing $x$ as a fixed constant, use the Fundamental Theorem of Calculus to evaluate the integral $$\begin{gathered} A(x) = \int_{-4}^4 f(x,y) \, dy. \end{gathered}$$ Note that you will be integrating with respect to $y$, and holding $x$ constant. Your result should be a function of $x$ only.

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

3. 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 this proposition.