Double Integrals over General Regions

Let $D$ be the region inside the unit circle centered at the origin, let $R$ be the right half of $D$, and let $B$ be the bottom half of $D$.

1. On three separate plots, graph and label the regions $D$, $R$, and $B$.

2. For each double integral below, decide without calculation whether the double integral is positive, negative, or zero. Write a sentence or two to explain your answer for each part.

   1. $\iint_D \; 1\; dA$
   2. $\iint_D \; x \; dA$
   3. $\iint_B \; x \; dA$
   4. $\iint_R \; x \; dA$
   5. $\iint_D \; x^2 \; dA$
   6. $\iint_B \; 2-x^2 \; dA$
   7. $\iint_R \; -e^x \; dA$
   8. $\iint_B \; 1-e^y \; dA$
   9. $\iint_D \; xy \; dA$

For each of the following iterated integrals,

• sketch the region of integration,

• write an equivalent iterated integral expression in the opposite order of integration,

• choose one of the two orders and evaluate the integral.

1. $\int_{x=0}^{x=1} \int_{y=x^2}^{y=x} xy \, dy \, dx$

2. $\int_{y=0}^{y=2} \int_{x=-\sqrt{4-y^2}}^{x=0} xy \, dx \, dy$

3. $\int_{x=0}^{x=1} \int_{y=x^4}^{y=x^{1/4}} x+y \, dy \, dx$

4. $\int_{y=0}^{y=2} \int_{x=y/2}^{x=2y} x+y \, dx \, dy$

Consider the solid that is given by the following description: the base is the given region $D$, while the top is given by the surface $z = p(x,y)$. In each setting below, set up, but do not evaluate, an iterated integral whose value is the exact volume of the solid. Include a labeled sketch of $D$ in each case.

1. $D$ is the interior of the quarter circle of radius 2, centered at the origin, that lies in the second q…