## Educators

ag

Problem 1

(a) Estimate the volume of the solid that lies below the surface $z = xy$ and above the rectangle
$$R = \{(x, y) \mid 0 \le x \le 6, 0 \le y \le 4 \}$$
Use a Riemann sum with $m = 3$, $n = 2$, and take the sample point to be the upper right corner of each square.
(b) Use the Midpoint Rule to estimate the volume of the solid in part (a).

ag
Alan G.

Problem 2

If $R = [0, 4] \times [-1, 2]$, use a Riemann sum with $m = 2$, $n = 3$ to estimate the value of $\iint_R (1 - xy^2)\ dA$. Take the sample points to be (a) the lower right corners and (b) the upper left corners of the rectangles.

ag
Alan G.

Problem 3

(a) Use a Riemann sum with $m = n = 2$ to estimate the value of $\iint_R xe^{-xy}\ dA$, where $R = [0, 2] \times [0, 1]$. Take the sample points to be upper right corners.
(b) Use the Midpoint Rule to estimate the integral in part (a).

ag
Alan G.

Problem 4

(a) Estimate the volume of the solid that lies below the surface $z = 1 + x^2 + 3y$ and above the rectangle $R = [1, 2] \times [0, 3]$. Use a Riemann sum with $m = n = 2$ and choose the sample points to be lower left corners.
(b) Use the Midpoint Rule to estimate the volume in part (a).

ag
Alan G.

Problem 5

Let $V$ be the volume of the solid that lies under the graph of $f(x, y) = \sqrt{52 - x^2 - y^2}$ and above the rectangle given by $2 \le x \le 4, 2 \le y \le 6$. Use the lines $x = 3$ and $y = 4$ to divide $R$ into subrectangles. Let $L$ and $U$ be the Riemann sums computed using lower left corners and upper right corners, respectively. Without calculating the numbers $V$, $L$, and $U$, arrange them in increasing order and explain your reasoning.

ag
Alan G.

Problem 6

A 20-ft-by-30-ft swimming pool is filled with water. The depth is measured at 5-ft intervals, starting at one corner of the pool, and the values are recorded in the table. Estimate the volume of water in the pool.

ag
Alan G.

Problem 7

A contour map is shown for a function $f$ on the square $R = [0, 4] \times [0, 4]$.

(a) Use the Midpoint Rule with $m = n = 2$ to estimate the value of $\iint_R f(x, y)\ dA$.
(b) Estimate the average value of $f$.

ag
Alan G.

Problem 8

The contour map shows the temperature, in degrees Fahrenheit, at 4:00 pm on February 26, 2007, in Colorado. (The state measures 388 mi west to east and 276 mi south to north.) Use the Midpoint Rule with $m = n = 4$ to estimate the average temperature in Colorado at that time.

ag
Alan G.

Problem 9

Evaluate the double integral by first identifying it as the volume of a solid.

$\iint_R \sqrt{2}\ dA$, $R = \{(x, y) \mid 2 \le x \le 6, -1 \le y \le 5 \}$

ag
Alan G.

Problem 10

Evaluate the double integral by first identifying it as the volume of a solid.

$\iint_R (2x + 1)\ dA$, $R = \{(x, y) \mid 0 \le x \le 2, 0 \le y \le 4 \}$

ag
Alan G.

Problem 11

Evaluate the double integral by first identifying it as the volume of a solid.

$\iint_R (4 - 2y)\ dA$, $R = [0, 1] \times [0, 1]$

ag
Alan G.

Problem 12

The integral $\iint_R \sqrt{9 - y^2}\ dA$, where $R = [0, 4] \times [0, 2]$, represents the volume of a solid. Sketch the solid.

ag
Alan G.

Problem 13

Find $\int_0^2 f(x, y)\ dx$ and $\int_0^3 f(x, y)\ dy$

$f(x, y) = x + 3x^2 y^2$

ag
Alan G.

Problem 14

Find $\int_0^2 f(x, y)\ dx$ and $\int_0^3 f(x, y)\ dy$

$f(x, y) = y \sqrt{x + 2}$

ag
Alan G.

Problem 15

Calculate the iterated integral.

$\displaystyle \int_1^4 \int_0^2 (6x^2y - 2x)\ dy dx$

ag
Alan G.

Problem 16

Calculate the iterated integral.

$\displaystyle \int_0^1 \int_0^1 (x + y)^2\ dx dy$

ag
Alan G.

Problem 17

Calculate the iterated integral.

$\displaystyle \int_0^1 \int_1^2 (x + e^{-y})\ dx dy$

ag
Alan G.

Problem 18

Calculate the iterated integral.

$\displaystyle \int_0^{\frac{\pi}{6}} \int_0^{\frac{\pi}{2}} (\sin x + \sin y)\ dy dx$

ag
Alan G.

Problem 19

Calculate the iterated integral.

$\displaystyle \int_{-3}^3 \int_0^{\frac{\pi}{2}} (y + y^2 \cos x)\ dx dy$

ag
Alan G.

Problem 20

Calculate the iterated integral.

$\displaystyle \int_1^3 \int_1^5 \frac{\ln y}{xy}\ dy dx$

ag
Alan G.

Problem 21

Calculate the iterated integral.

$\displaystyle \int_1^4 \int_1^2 \left (\frac{x}{y} + \frac{y}{x} \right)\ dy dx$

ag
Alan G.

Problem 22

Calculate the iterated integral.

$\displaystyle \int_0^1 \int_0^2 ye^{x - y}\ dx dy$

ag
Alan G.

Problem 23

Calculate the iterated integral.

$\displaystyle \int_0^3 \int_0^{\frac{\pi}{2}} t^2 \sin^3 \phi d \phi\ dt$

ag
Alan G.

Problem 24

Calculate the iterated integral.

$\displaystyle \int_0^1 \int_0^1 xy \sqrt{x^2 + y^2}\ dy dx$

ag
Alan G.

Problem 25

Calculate the iterated integral.

$\displaystyle \int_0^1 \int_0^1 v(u + v^2)^4\ du dv$

ag
Alan G.

Problem 26

Calculate the iterated integral.

$\displaystyle \int_0^1 \int_0^1 \sqrt{s + t}\ ds dt$

ag
Alan G.

Problem 27

Calculate the double integral.

$\displaystyle \iint\limits_R x \sec^2 y\ dA$, $R = \{(x, y) \mid 0 \le x \le 2, 0 \le y \le \frac{\pi}{4} \}$

ag
Alan G.

Problem 28

Calculate the double integral.

$\displaystyle \iint\limits_R (y + xy^{-2})\ dA$, $R = \{(x, y) \mid 0 \le x \le 2, 1 \le y \le 2 \}$

ag
Alan G.

Problem 29

Calculate the double integral.

$\displaystyle \iint\limits_R \frac{xy^2}{x^2 + 1}\ dA$, $R = \{(x, y) \mid 0 \le x \le 1, -3 \le y \le 3 \}$

ag
Alan G.

Problem 30

Calculate the double integral.

$\displaystyle \iint\limits_R \frac{\tan \theta}{\sqrt{1 - t^2}}\ dA$, $R = \{(\theta, t) \mid 0 \le \theta \le \frac{\pi}{3}, 0 \le t \le \frac{1}{2} \}$

ag
Alan G.

Problem 31

Calculate the double integral.

$\displaystyle \iint\limits_R x \sin (x + y)\ dA$, $R = [0, \frac{\pi}{6}] \times [0, \frac{\pi}{3}]$

ag
Alan G.

Problem 32

Calculate the double integral.

$\displaystyle \iint\limits_R \frac{x}{1 + xy}\ dA$, $R = [0, 1] \times [0, 1]$

ag
Alan G.

Problem 33

Calculate the double integral.

$\displaystyle \iint\limits_R ye^{-xy}\ dA$, $R = [0, 2] \times [0, 3]$

ag
Alan G.

Problem 34

Calculate the double integral.

$\displaystyle \iint\limits_R \frac{1}{1 + x + y}\ dA$, $R = [1, 3] \times [1, 2]$

ag
Alan G.

Problem 35

Sketch the solid whose volume is given by the iterated integral.

$\displaystyle \int_0^1 \int_0^1 (4 - x - 2y)\ dx dy$

ag
Alan G.

Problem 36

Sketch the solid whose volume is given by the iterated integral.

$\displaystyle \int_0^1 \int_0^1 (2 - x^2 - y^2 )\ dy dx$

ag
Alan G.

Problem 37

Find the volume of the solid that lies under the plane $4x + 6y - 2z + 15 = 0$ and above the rectangle $R = \{(x, y) \mid -1 \le x \le 2, -1 \le y \le 1\}$.

ag
Alan G.

Problem 38

Find the volume of the solid that lies under the hyperbolic paraboloid $z = 3y^2 - x^2 + 2$ and above the rectangle $R = [-1, 1] \times [1, 2]$.

ag
Alan G.

Problem 39

Find the volume of the solid lying under the elliptic paraboloid $\frac{x^2}{4} + \frac{y^2}{9} + z = 1$ and above the rectangle $R = [-1, 1] \times [-2, 2]$.

ag
Alan G.

Problem 40

Find the volume of the solid enclosed by the surface $z = x^2 + xy^2$ and the planes $z = 0$, $x = 0$, $x = 5$, and $y = \pm 2$.

ag
Alan G.

Problem 41

Find the volume of the solid enclosed by the surface $z = 1 + x^2 ye^y$ and the planes $z = 0$, $x = \pm 1$, $y = 0$, and $y = 1$.

ag
Alan G.

Problem 42

Find the volume of the solid in the first octant bounded by the cylinder $z = 16 - x^2$ and the plane $y = 5$.

ag
Alan G.

Problem 43

Find the volume of the solid enclosed by the paraboloid $z = 2 + x^2 + (y - 2)^2$ and the planes $z = 1$, $x = 1$, $x = -1$, $y = 0$, and $y = 4$.

ag
Alan G.

Problem 44

Graph the solid that lies between the surface $z = \frac{2xy}{(x^2 + 1)}$ and the plane $z = x + 2y$ and is bounded by the planes $x = 0$, $x = 2$, $y = 0$, and $y = 4$. Then find its volume.

ag
Alan G.

Problem 45

Use a computer algebra system to find the exact value of the integral $\int \int_R x^5 y^3 e^{xy}\ dA$, where $R = [0, 1] \times [0, 1]$. Then use the $CAS$ to draw the solid whose volume is given by the integral.

ag
Alan G.

Problem 46

Graph the solid that lies between the surfaces $z = e^{-x^2} \cos (x^2 + y^2)$ and $z = 2 - x^2 - y^2$ for $\mid x \mid \le 1$, $\mid y \mid \le 1$. Use a computer algebra system to approximate the volume of this solid correct to four decimal places.

ag
Alan G.

Problem 47

Find the average value of $f$ over the given rectangle.

$f(x, y) = x^2 y$,
$R$ has vertices $(-1, 0)$, $(-1, 5)$, $(1, 5)$, $(1, 0)$

ag
Alan G.

Problem 48

Find the average value of $f$ over the given rectangle.

$f(x, y) = e^y \sqrt{x + e^y}$, $R = [0, 4] \times [0, 1]$

ag
Alan G.

Problem 49

Use symmetry to evaluate the double integral.

$\displaystyle \iint\limits_R \frac{xy}{1 + x^4}\ dA$, $R = \{(x, y) \mid -1 \le x \le 1, 0 \le y \le 1 \}$

ag
Alan G.

Problem 50

Use symmetry to evaluate the double integral.

$\displaystyle \iint\limits_R (1 + x^2 \sin y + y^2 \sin x)\ dA$, $R = [-\pi, \pi] \times [-\pi, \pi]$

ag
Alan G.

Problem 51

Use a $CAS$ to compute the iterated integrals

$\displaystyle \int_0^1 \int_0^1 \frac{x - y}{(x + y)^3}\ dy dx$ and $\displaystyle \int_0^1 \int_0^1 \frac{x - y}{(x + y)^3}\ dx dy$

ag
Alan G.

Problem 52

(a) In what way are the theorems of Fubini and Clairaut similar?
(b) If $f(x, y)$ is continuous on $[a, b] \times [c, d]$ and
$$g(x, y) = \int_a^x \int_c^y f(s, t)\ dt ds$$
for $a < x < b, c < y < d$, show that $g_{xy} = g_{yx} = f(x, y)$.

ag
Alan G.