True or False $\int_{0}^{2}\big[\int_{0}^{1}xy^{2}{\it dy}\big] {\it dx}=\big[\int_{0}^{2}x\,{\it dx}\big]\cdot \big[\int_{0}^{1}y^{2}{\it dy}\big]$.

True or False Fubini’s Theorem states that if a function $z=f(x,y)$ is continuous on a closed rectangular region $R$ defined by $a\leq x\leq b$ and $c\leq y\leq d$, then $\int_{c}^{d} \big[ \int_{a}^{b}f(x,y)\,{\it dx}\big] {\it dy}=\int_{a}^{b}\big[ \int_{c}^{d}f(x,y)\,{\it dx}\big] {\it dy}.$

Multiple Choice The result of integrating $\int_{1}^{4}x^{2}\sqrt{y}\,{\it dy}$ is [(a) a number, (b) a function of $y,$ (c) a function of $x$ and $y$, (d) a function of $x$].

True or False If a function $z=f(x,y)$ is continuous on a closed, rectangular region $R,$ then the double integral $\displaystyle\iint\limits_{\kern-3ptR}f(x,y)\,{\it dA}$ exists.

Let $f(x,y) =x(3-y)$ be defined over the region $R$ shown in the figure.

Let $f(x,y) =3xy^{2}$ be defined over the rectangular region $R$ defined by $0\leq x\leq 6,$ $0\leq y\leq 4.$

Let $f(x,y) =x^{2}+y$ be defined over the rectangular region $R$ defined by $1\leq x\leq 5,$ $2\leq y\leq 4.$

Let $f(x,y) =x\left( 1-y\right)$ be defined over the rectangular region $R$ defined by $0\leq x\leq 3,$ $0\leq y\leq 4.$

Find the Riemann sum for Example 1 if $R$ is divided into four congruent subsquares. Choose $(u_{k},v_{k}),$ $k=1,2,3,4,$ as the center of the $k$th subsquare.

Find the Riemann sum for Example 1 if $R$ is divided into eight congruent subrectangles with $\Delta x_{i}=1,$ $i=1,2,3,4$, and $\Delta y_{j}=2,$ $j=1,2$. Choose $(u_{k},v_{k}),$ $k=1,2,3,\ldots ,8,$ as the center of the $k$th subrectangle.

$\int_{1}^{e}\dfrac{x}{y}dy$

$\int_{0}^{2}\dfrac{x}{y}{\it dx}$

$\int_{0}^{\pi /2}x\sin y\,{\it dx}$

$\int_{0}^{\pi /2}x\sin y\,{\it dy}$

$\int_{0}^{1}e^{x}\,{\it dx}$

$\int_{0}^{1}e^{x}\,{\it dy}$

$\displaystyle\int_{0}^{1}\left[ \int_{0}^{2}x^{2}y\,{\it dy}\right] \! {\it dx}$

$\displaystyle\int_{0}^{2}\left[ \int_{0}^{4}x^{2}y\,{\it dx} \right] \! {\it dy}$

$\displaystyle\int_{0}^{3}\left[ \int_{0}^{2}3xy\,{\it dy}\right] \! {\it dx}$

$\displaystyle\int_{0}^{2}\left[ \int_{0}^{1}3xy\,{\it dx}\right] \! {\it dy}$

$\displaystyle\int_{-1}^{1}\left[ \int_{0}^{1}3x^{2}y^{2}\,{\it dx} \right] \! {\it dy}$

$\displaystyle\int_{0}^{1}\left[\int_{0}^{2}3x^{2}y^{2}\,{\it dy}\right] \! {\it dx}$

$\displaystyle\int_{0}^{2}\left[ \int_{-1}^{1}2xy^{2}\,{\it dx}\right] \! {\it dy}$

$\displaystyle\int_{-1}^{1}\left[ \int_{1}^{2}2xy^{2}\,{\it dy} \right] \! {\it dx}$

$\displaystyle\int_{0}^{\pi /4}\left[\int_{0}^{2}x\cos y\,{\it dx}\right] \! {\it dy}$

$\displaystyle\int_{0}^{2}\left[ \int_{0}^{\pi /3}x\cos y\,{\it dy}\right] \! {\it dx}$

$\displaystyle\int_{0}^{3}\left[ \int_{0}^{\pi /3} (4x-3) ^{2}\sin y\,{\it dy}\right]\! {\it dx}$

$\displaystyle\int_{0}^{\pi /3}\left[ \int_{0}^{3}(4x-3) \sin y\,{\it dx}\right] \! {\it dy}$

$\displaystyle\int_{0}^{1}\left[ \int_{0}^{\pi /2}e^{x}\cos y\,{\it dy} \right]\! {\it dx}$

$\displaystyle\int_{0}^{\pi /4}\left[ \int_{0}^{1}e^{x}\cos y\,{\it dx}\right]\! {\it dy}$

$\displaystyle\int_{0}^{2}\left[ \int_{-\pi /2}^{\pi /2}\dfrac{\sin y}{2x+1}\,{\it dy}\right]\! {\it dx}$

$\displaystyle\int_{0}^{\pi /2}\left[ \int_{0}^{2}\dfrac{\sin y}{2x+1} \,{\it dx}\right]\! {\it dy}$

$\displaystyle\int_{0}^{\pi/2}\left[ \int_{0}^{2}xe^{x}\sin y\,{\it dx}\right]\! {\it dy}$

$\displaystyle\int_{0}^{1}\left[ \int_{0}^{\pi/2}xe^{x}\sin y\,{\it dy}\right]\! {\it dx}$

$\displaystyle\int_{1}^{2}\left[ \int_{0}^{\pi /2}\dfrac{x\cos x}{y}\,{\it dx} \right]\! {\it dy}$

$\displaystyle\int_{0}^{\pi /3}\left[ \int_{1}^{2}\dfrac{x\cos x}{y}\,{\it dy}\right]\! {\it dx}$

$\displaystyle\iint\limits_{\kern-3ptR} ( 2x+y^{2})\, {\it dA}, \quad 0\leq x\leq 2,$ $0\leq y\leq 3$

$\displaystyle\iint\limits_{\kern-3ptR}( x^{2}-3y)\, {\it dA}, \quad -1\leq x\leq 2,$ $0\leq y\leq 1$

$\displaystyle\iint\limits_{\kern-3ptR}x( x^{2}+5) \, {\it dA}, \quad 0\leq x\leq 2,$ $-1\leq y\leq 1$

$\displaystyle\iint\limits_{\kern-3ptR}\sqrt{y}( 3x^{2}+x)\, {\it dA}, \quad 0\leq x\leq 2,$ $0\leq y\leq 4$

$\displaystyle\iint\limits_{\kern-3ptR}2xe^{y}{\it dA}, \quad -3\leq x\leq 2,$ $0\leq y\leq 1$

$\displaystyle\iint\limits_{\kern-3ptR}e^{2x+y}{\it dA}, \quad 0\leq x\leq 1,$ $-1\leq y\leq 1$

$\displaystyle\iint\limits_{\kern-3ptR}x\sec ^{2}y\,{\it dA}, \quad 0\leq x\leq 3,$ $0\leq y\leq \dfrac{\pi }{4}$

$\displaystyle\iint\limits_{R}y^{2}\sec x\tan x\,{\it dA}, \quad 0\leq x\leq \dfrac{\pi }{3},$ $-1\leq y\leq 3$

$\displaystyle\iint\limits_{\kern-3ptR}\dfrac{x}{2y+3}\,{\it dA}, \quad 0\leq x\leq 2,$ $0\leq y\leq 1$

$\displaystyle\iint\limits_{\kern-3ptR}\dfrac{y^{2}}{x-3}\,{\it dA}, \quad 4\leq x\leq 5,$ $0\leq y\leq 3$

$\displaystyle\iint\limits_{\kern-3ptR}x\sin (xy)\, {\it dA}, \quad 0\leq x\leq \dfrac{\pi }{2}$, $0\leq y\leq 1$

$\displaystyle\iint\limits_{\kern-3ptR}y\cos (xy)\, {\it dA}, \quad 0\leq x\leq 1$, $\dfrac{\pi }{2}\leq y\leq 2\pi$

$\displaystyle\iint\limits_{\kern-3ptR}x^{3}\cos ( x^{2}y)\, {\it dA}, \quad 0\leq x\leq \dfrac{\pi }{2}$, $0\leq y\leq 1$

$\displaystyle\iint\limits_{\kern-3ptR}x^{3}\sin ( x^{2}y)\, {\it dA}, \quad 0\leq x\leq \dfrac{\pi }{2}$, $0\leq y\leq 1$

$\displaystyle\iint\limits_{\kern-3ptR}\dfrac{y^{2}}{(1+xy^{2}) ^{3}}\,{\it dA}, \quad 0\leq x\leq 2,$ $0\leq y\leq 3$

$\displaystyle\iint\limits_{\kern-3ptR}( x^{3}+x) ( x^{2}y+y)\, {\it dA}, \quad 0\leq x\leq 1,$ $0\leq y\leq 1$

$\displaystyle\iint\limits_{\kern-3ptR}y^{5}e^{xy^{3}}{\it dA}, \quad 0\leq x\leq 1,$ $0\leq y\leq 2$

$\displaystyle\iint\limits_{\kern-3ptR}x^{5}ye^{x^{3}y^{2}}{\it dA}, \quad 0\leq x\leq 2,$ $0\leq y\leq 1$

$f(x,y)=x+2y, \quad 0\leq x\leq 1,$ $0\leq y\leq 2$

$f(x,y)=2x+3y, \quad 0\leq x\leq 2,$ $0\leq y\leq 3$

$f(x,y)=x^{2}+y^{2}, \quad 0\leq x\leq 2,$ $0\leq y\leq 1$

$f(x,y)=4x^{2}+3y^{2}, \quad 0\leq x\leq 3,$ $-2\leq y\leq 1$

$f(x,y)=\sin x, \quad 0\leq x\leq \dfrac{\pi }{2},$ $0\leq y\leq 1$

$f(x,y)=\cos y, \quad 0\leq x\leq 1,$ $0\leq y\leq \dfrac{\pi }{2}$

$f(x,y)=\dfrac{x^{2}+y^{2}}{xy} \quad 1\leq x\leq 2,$ $1\leq y\leq 2$

$f(x,y)=\dfrac{y^{2}}{x^{2}}, \quad 1\leq x\leq 2,$ $1\leq y\leq 2$

$f(x,y)=e^{x+y}, \quad 0\leq x\leq 2,$ $0\leq y\leq 1$

$f(x,y)=e^{x-y}, \quad 0\leq x\leq 2,$ $1\leq y\leq 2$

Volume Find the volume of the solid below the paraboloid $z=x^{2}+y^{2}$ and above the square in the $xy$-plane enclosed by the lines $x=\pm 1$ and $y=\pm 1$.

Show that $\displaystyle\iint\limits_{\kern-3ptR}{\it dA}=\int_{c}^{d}\big[ \int_{a}^{b}{\it dx}\big] {\it dy} =(b-a) (d-c)$. That is, show that the volume of a solid with height $1$, $\displaystyle\iint\limits_{\kern-3ptR}{\it dA},$ defined over a rectangular region $a\leq x\leq b,$ $c\leq y\leq d$ is numerically equal to the area of the rectangle.

Average Value of a Function Suppose that $z=f(x,y)$ is integrable over a closed, rectangular region $R$ in the $xy$-plane. Let $P$ be a partition of $R$ into $N$ subrectangles of equal area $\Delta A$. Evaluate $f$ at the center $(u_{k},v_{k})$ of the $k$th subrectangle $(k=1,2,\ldots ,N)$, and let AVG be the average of these $N$ values.

$f(x,y) =y\cos x, \quad 0\leq x\leq \pi ,$ $1\leq y\leq 5$

$f(x,y) =\dfrac{xy}{x^{2}+1}, \quad 0\leq x\leq 1,$ $0\leq y\leq 2$