Iterated Integrals

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 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…

Let $f(x,y) = x+y^2$ on the rectangle $R$ with $0\leq x\leq 2$ and $0\leq y\leq 3$.

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 $y$.

2. 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).

Evaluate each of the following double or iterated integrals exactly.

1. $\int_1^3 \left( \int_2^5 xy \, dy \right) \, dx$

2. $\int_0^{\pi/4} \left( \int_0^{\pi/3} \sin(x) \cos(y) \, dx \right) \, dy$

3. $\int_0^1 \left( \int_0^1 e^{-2x - 3y} \, dy \right) \, dx$

4. $\iint_R \sqrt{2x + 5y} \, dA$, where $R = [0,2]\times[0,3]$.

The temperature at any point on a metal plate in the $xy$ plane is given by $T(x,y) = 100-4x^2 - y^2$, where $x$ and $y$ are measured in inches and $T$ in degrees Celsius. Consider the portion of the plate that lies on the rectangular region $R = [1,5] \times [3,6]$.

1. Write an iterated integral whose value represents the volume under the surface $T$ over the rectangle $R$.

2. Evaluate the iterated integral you determined in (a).

3. Find the area of the rectangle, $R$.

4. …

Consider the box with a sloped top that is given by the following description: the base is the rectangle $R = [1,4] \times [2,5]$, while the top is given by the plane $z = p(x,y) = 30 - x - 2y$.

1. Write an iterated integral whose value represents the volume under $p$ over the rectangle $R$.

2. Evaluate the iterated integral you determined in (a).

3. What is the exact average value of $p$ over $R$?

4. If you wanted to build a rectangular box (with an identical base) t…