Section 1
Double and Iterated Integrals over Rectangles
In Exercises $1-14,$ evaluate the iterated integral.$$\int_{1}^{2} \int_{0}^{4} 2 x y d y d x$$
In Exercises $1-14,$ evaluate the iterated integral.$$\int_{0}^{2} \int_{-1}^{1}(x-y) d y d x$$
In Exercises $1-14,$ evaluate the iterated integral.$$\int_{-1}^{0} \int_{-1}^{1}(x+y+1) d x d y$$
In Exercises $1-14,$ evaluate the iterated integral.$$\int_{0}^{1} \int_{0}^{1}\left(1-\frac{x^{2}+y^{2}}{2}\right) d x d y$$
In Exercises $1-14,$ evaluate the iterated integral.$$\int_{0}^{3} \int_{0}^{2}\left(4-y^{2}\right) d y d x$$
In Exercises $1-14,$ evaluate the iterated integral.$$\int_{0}^{3} \int_{-2}^{0}\left(x^{2} y-2 x y\right) d y d x$$
In Exercises $1-14,$ evaluate the iterated integral.$$\int_{0}^{1} \int_{0}^{1} \frac{y}{1+x y} d x d y$$
In Exercises $1-14,$ evaluate the iterated integral.$$\int_{1}^{4} \int_{0}^{4}\left(\frac{x}{2}+\sqrt{y}\right) d x d y$$
In Exercises $1-14,$ evaluate the iterated integral.$$\int_{0}^{\ln 2} \int_{1}^{\ln 5} e^{2 x+y} d y d x$$
In Exercises $1-14,$ evaluate the iterated integral.$$\int_{0}^{1} \int_{1}^{2} x y e^{x} d y d x$$
In Exercises $1-14,$ evaluate the iterated integral.$$\int_{-1}^{2} \int_{0}^{\pi / 2} y \sin x d x d y$$
In Exercises $1-14,$ evaluate the iterated integral.$$\int_{\pi}^{2 \pi} \int_{0}^{\pi}(\sin x+\cos y) d x d y$$
In Exercises $1-14,$ evaluate the iterated integral.$$\int_{1}^{4} \int_{1}^{e} \frac{\ln x}{x y} d x d y$$
In Exercises $1-14,$ evaluate the iterated integral.$$\int_{-1}^{2} \int_{1}^{2} x \ln y d y d x$$
Find all values of the constant $c$ so that $\int_{0}^{1} \int_{0}^{c}(2 x+y) d x d y=3$
In Exercises $17-24$ , evaluate the double integral over the given region $R .$Find all values of the constant $c$ so that$$\int_{-1}^{c} \int_{0}^{2}(x y+1) d y d x=4+4 c$$
In Exercises $17-24$ , evaluate the double integral over the given region $R .$$$\iint_{R}\left(6 y^{2}-2 x\right) d A, \quad R : 0 \leq x \leq 1, \quad 0 \leq y \leq 2$$
In Exercises $17-24$ , evaluate the double integral over the given region $R .$$$\iint_{R}\left(\frac{\sqrt{x}}{y^{2}}\right) d A, \quad R : \quad 0 \leq x \leq 4, \quad 1 \leq y \leq 2$$
In Exercises $17-24$ , evaluate the double integral over the given region $R .$$$\iint_{R} x y \cos y d A, \quad R :-1 \leq x \leq 1, \quad 0 \leq y \leq \pi$$
In Exercises $17-24$ , evaluate the double integral over the given region $R .$$$\iint_{R} y \sin (x+y) d A, \quad R : \quad-\pi \leq x \leq 0, \quad 0 \leq y \leq \pi$$
In Exercises $17-24$ , evaluate the double integral over the given region $R .$$$\iint_{R} e^{x-y} d A, \quad R : \quad 0 \leq x \leq \ln 2, \quad 0 \leq y \leq \ln 2$$
In Exercises $17-24$ , evaluate the double integral over the given region $R .$$$\iint_{R} x y e^{\mathrm{y}^{2}} d A, \quad R : \quad 0 \leq x \leq 2, \quad 0 \leq y \leq 1$$
In Exercises $17-24$ , evaluate the double integral over the given region $R .$$$\iint_{R} \frac{x y^{3}}{x^{2}+1} d A, \quad R : \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 2$$
In Exercises $17-24$ , evaluate the double integral over the given region $R .$$$\iint_{R} \frac{y}{x^{2} y^{2}+1} d A, \quad R : \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 1$$
In Exercises 25 and $26,$ integrate $f$ over the given region.$$\begin{array}{l}{\text { Square } f(x, y)=1 /(x y) \quad \text { over } \quad \text { the square } \quad 1 \leq x \leq 2} \\ {1 \leq y \leq 2}\end{array}$$
Rectangle $f(x, y)=y \cos x y$ over the rectangle $0 \leq x \leq \pi$$0 \leq y \leq 1$
In Exercises 27 and $28,$ sketch the solid whose volume is given by thespecified integral.$$\int_{0}^{1} \int_{0}^{2}\left(9-x^{2}-y^{2}\right) d y d x$$
In Exercises 27 and $28,$ sketch the solid whose volume is given by thespecified integral.$$\int_{0}^{3} \int_{1}^{4}(7-x-y) d x d y$$
Find the volume of the region bounded above by the paraboloid $z=x^{2}+y^{2}$ and below by the square $R :-1 \leq x \leq 1$ $-1 \leq y \leq 1 .$
Find the volume of the region bounded above by the elliptical paraboloid $z=16-x^{2} y^{2}$ and below by the square $R : 0 \leq x \leq 2,0 \leq y \leq 2$
Find the volume of the region bounded above by the plane $z=2-x-y$ and below by the square $R : 0 \leq x \leq 1$ $0 \leq y \leq 1$
Find the volume of the region bounded above by the plane $z=y / 2$ and below by the rectangle $R : 0 \leq x \leq 4,0 \leq y \leq 2$
Find the volume of the region bounded above by the surface $z=2 \sin x \cos y$ and below by the rectangle $R : 0 \leq x \leq \pi / 2$ $0 \leq y \leq \pi / 4$
Find the volume of the region bounded above by the surface $z=4-y^{2}$ and below by the rectangle $R : 0 \leq x \leq 1$ $0 \leq y \leq 2$
Find a value of the constant $k$ so that $\int_{1}^{2} \int_{0}^{3} k x^{2} y d x d y=1$
Evaluate $\int_{-1}^{1} \int_{0}^{\pi / 2} x \sin \sqrt{y} d y d x$
Use Fubini's Theorem to evaluate $$\int_{0}^{2} \int_{0}^{1} \frac{x}{1+x y} d x d y$$
Use Fubini's Theorem to evaluate$$\int_{0}^{1} \int_{0}^{3} x e^{x y} d x d y$$
Use a software application to compute the integrals $$ \text a.\int_{0}^{1} \int_{0}^{2} \frac{y-x}{(x+y)^{3}} d x d y \quad b. \int_{0}^{2} \int_{0}^{1} \frac{y-x}{(x+y)^{3}} d y d x$$ Explain why your results do not contradict Fubini's Theorem.
If $f(x, y)$ is continuous over $R : a \leq x \leq b, c \leq y \leq d$ and $$F(x, y)=\int_{a}^{x} \int_{c}^{y} f(u, v) d v d u$$ on the interior of $R,$ find the second partial derivatives $F_{x y}$ and $F_{y x}$