Question 1

Find a region R such that double integral $\int _ { R } \left( 2 - x ^ { 2 } - y ^ { 2 } \right) d A$ has the largest value.

Accepted Answer

A)  $x ^ { 2 } - y ^ { 2 } \leq 2$ 
B)  $x ^ { 2 } - y ^ { 2 } \geq 2$ 
C)  $x ^ { 2 } + y ^ { 2 } \leq 2$ 
D)  $x ^ { 2 } + y ^ { 2 } \geq 2$ 
A)  $x ^ { 2 } - y ^ { 2 } \leq 2$ 
B)  $x ^ { 2 } - y ^ { 2 } \geq 2$ 
C)  $x ^ { 2 } + y ^ { 2 } \leq 2$ 
D)  $x ^ { 2 } + y ^ { 2 } \geq 2$

Question 2

Consider the change of variables x = s + 5t, y = s - 3t.
Find the absolute value of the Jacobian $\left| \frac { \partial ( x , y ) } { \partial ( s , t ) } \right|$ .

Accepted Answer

The answer of Consider the change of variables x =...

Question 3

Let W be the region between the cylinders $x ^ { 2 } + y ^ { 2 } = 9$ and $x ^ { 2 } + y ^ { 2 } = 36$ in the first octant and under the plane z = 1.Evaluate $\int _ { W } Z d V$

Accepted Answer

The answer of Let W be the region between the...

Question 4

Set up but do not evaluate a (multiple)integral that gives the volume of the solid bounded above by the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 2$ and below by the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ .

Accepted Answer

A)  $\int _ { - 1 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { x ^ { 2 } + y ^ { 2 } } ^ { \sqrt { 2 - x ^ { 2 } - y ^ { 2 } } } d x d y d z$ 
B)  $\int _ { - 1 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { - x ^ { 2 } - y ^ { 2 } } ^ { \sqrt { 2 - x ^ { 2 } - y ^ { 2 } } } d z d y d x$ 
C)  $$
\int_{-1}^{1} \int_{0}^{\sqrt{1-x^{2}}} \int_{x^{2}+y^{2}}^{\sqrt{2-x^{2}-y^{2}}}$$$ d z d y d x $ 
D)  $\int _ { - 1 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { x ^ { 2 } + y ^ { 2 } } ^ { \sqrt { 2 - x ^ { 2 } - y ^ { 2 } } } d z d y d x$ 
E)  $\int _ { 0 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { x ^ { 2 } + y ^ { 2 } } ^ { \sqrt { 2 - x ^ { 2 } - y ^ { 2 } } } d z d y d x$ 
A)  $\int _ { - 1 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { x ^ { 2 } + y ^ { 2 } } ^ { \sqrt { 2 - x ^ { 2 } - y ^ { 2 } } } d x d y d z$ 
B)  $\int _ { - 1 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { - x ^ { 2 } - y ^ { 2 } } ^ { \sqrt { 2 - x ^ { 2 } - y ^ { 2 } } } d z d y d x$ 
C)  $$
\int_{-1}^{1} \int_{0}^{\sqrt{1-x^{2}}} \int_{x^{2}+y^{2}}^{\sqrt{2-x^{2}-y^{2}}}$$$ d z d y d x $ 
D)  $\int _ { - 1 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { x ^ { 2 } + y ^ { 2 } } ^ { \sqrt { 2 - x ^ { 2 } - y ^ { 2 } } } d z d y d x$ 
E)  $\int _ { 0 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { x ^ { 2 } + y ^ { 2 } } ^ { \sqrt { 2 - x ^ { 2 } - y ^ { 2 } } } d z d y d x$

Question 5

Consider the integral $\int _ { - 6 } ^ { 6 } \int _ { 6 } ^ { \sqrt { 72 - x ^ { 2 } } } f ( x , y ) d y d x$ .Convert the integral to polar coordinates.

Accepted Answer

The answer of Consider the integral \(\int _ { -...

Question 6

Evaluate the integral by interchanging the order of integration. $\int _ { - 1 } ^ { 0 } \int _ { - 4 x } ^ { 4 } e ^ { y ^ { 2 } } d y d x$ .

Accepted Answer

The answer of Evaluate the integral by interchanging the order...

Question 7

Rewrite the integral $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { \sqrt { 4 - y ^ { 2 } } } \int _ { \sqrt { x ^ { 2 } + y ^ { 2 } } } ^ { \sqrt { 8 - x ^ { 2 } - y ^ { 2 } } } \left( x ^ { 2 } + y ^ { 2 } \right) d z d x d y$ in spherical coordinates.You do not have to evaluate the integral.

Accepted Answer

The answer of Rewrite the integral \(\int _ { 0...

Question 8

If f and g are two continuous functions on a region R, then $\int _ { R } f \cdot g d A = \int _ { R } f d A \cdot \int _ { R } g d A$ .

Accepted Answer

A) True 
 B)False

Question 9

For the following region, decide whether to integrate using polar or Cartesian coordinates.Write an iterated integral of an arbitrary function f(x, y)over the region.

Accepted Answer

The answer of For the following region, decide whether to...

Question 10

Evaluate the integral $\int _ { x } x + 3 y d A$ , where R is the region in the first quadrant bounded by the y-axis, the line y = x and the circles $x ^ { 2 } + y ^ { 2 } = 25 , x ^ { 2 } + y ^ { 2 } = 16$

Accepted Answer

The answer of Evaluate the integral \(\int _ { x...

Question 11

Convert the integral $\int _ { - 3 } ^ { 3 } \int _ { 0 } ^ { \sqrt { 9 - x ^ { 2 } } } e ^ { - \left( x ^ { 2 } + y ^ { 2 } \right) } d y d x$ to polar coordinates and hence evaluate it exactly.

Accepted Answer

The answer of Convert the integral \(\int _ { -...

Question 12

Let W be the part of the solid sphere of radius 4, centered at the origin, that lies above the plane z = 2.Express $\int _ { W } z d V$ in
(a)Cartesian
(b)Cylindrical
(c)Spherical coordinates.

Accepted Answer

(a)Cartesian coordinates.Integrate first

Question 13

The joint density function for x, y is given by $p ( x , y ) = \left\{ \begin{array} { c c } 
\frac { 1 } { 150 } e ^ { - x / 10 } e ^ { - y / 15 } & x \geq 0 , y \geq 0 \
0 & \text { otherwise }
\end{array} \right.$ Write down an iterated integral to compute the probability that x + y $\le$ 10.You do not need to do the integral.

Accepted Answer

A)  $\int _ { 0 } ^ { 10 } \int _ { 0 } ^ { 10 } \frac { 1 } { 150 } e ^ { - x / 10 } e ^ { - y / 15 } d ^ { } y d x$ 
B)  $\int _ { 0 } ^ { 10 } \int _ { 0 } ^ { 10 - x } \frac { 1 } { 150 } e ^ { - x / 10 } e ^ { - y / 15 } d x d y$ 
C)  $\int _ { 0 } ^ { 10 } \int _ { 0 } ^ { 10 - x } \frac { 1 } { 150 } e ^ { - x / 10 } e ^ { - y / 15 } d y d x$ 
D)  $\int _ { 0 } ^ { 10 } \int _ { 0 } ^ { 10 - x } \frac { 1 } { 10 } e ^ { - x / 10 } e ^ { - y / 15 } d y d ^ { 2 } x$ 
E)  $\int _ { 0 } ^ { 10 - x } \int _ { 0 } ^ { 10 } \frac { 1 } { 150 } e ^ { - x / 10 } e ^ { - y / 15 } d x d y$ 
A)  $\int _ { 0 } ^ { 10 } \int _ { 0 } ^ { 10 } \frac { 1 } { 150 } e ^ { - x / 10 } e ^ { - y / 15 } d ^ { } y d x$ 
B)  $\int _ { 0 } ^ { 10 } \int _ { 0 } ^ { 10 - x } \frac { 1 } { 150 } e ^ { - x / 10 } e ^ { - y / 15 } d x d y$ 
C)  $\int _ { 0 } ^ { 10 } \int _ { 0 } ^ { 10 - x } \frac { 1 } { 150 } e ^ { - x / 10 } e ^ { - y / 15 } d y d x$ 
D)  $\int _ { 0 } ^ { 10 } \int _ { 0 } ^ { 10 - x } \frac { 1 } { 10 } e ^ { - x / 10 } e ^ { - y / 15 } d y d ^ { 2 } x$ 
E)  $\int _ { 0 } ^ { 10 - x } \int _ { 0 } ^ { 10 } \frac { 1 } { 150 } e ^ { - x / 10 } e ^ { - y / 15 } d x d y$

Question 14

Evaluate the integral $\int _ { 0 } ^ { 2 } \int _ { - \sqrt { 4 - x ^ { 2 } } } ^ { 0 } \int _ { - \sqrt { 4 - x ^ { 2 } - y ^ { 2 } } } ^ { \sqrt { 4 - x ^ { 2 } - y ^ { 2 } } } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } d z d y d x$ in spherical coordinates.

Accepted Answer

The answer of Evaluate the integral \(\int _ { 0...

Question 15

Let R1 be the region 0 $\le$ x $\le$ 3, -2 $\le$ y $\le$ 4, and let R2 be the region 3 $\le$ x$\le$5, -2 $\le$ y $\le$ 4.Suppose that the average value of f over R1 is 6 and the average value over R2 is 7.
Find the average value of f over R, $0 \leq x \leq 5 , - 2 \leq y \leq 2$ .

Accepted Answer

The answer of Let R1 be the region 0 $\le$...

Question 16

Consider the integral $\int _ { 1 } ^ { 5 } \int _ { 9 x } ^ { 45 } f ( x , y ) d y d x$ Rewrite the integral with the integration performed in the opposite order.

Accepted Answer

A)  $\int _ { 9 x } ^ { A 5 } \int _ { 1 } ^ { y / 9 } f ( x , y ) d x d y$ 
B)  $\int _ { 9 } ^ { 4 5 } \int _ { 1 } ^ { y / 9 } f ( x , y ) d y d x$ 
C)  $\int _ { 9 } ^ { 4 5 } \int _ { 1 } ^ { y / 9 } f ( x , y ) d x d y$ 
D)  $\int _ { 9 } ^ { 45 } \int _ { 1 } ^ { 0 y } f ( x , y ) d x d y$ 
A)  $\int _ { 9 x } ^ { A 5 } \int _ { 1 } ^ { y / 9 } f ( x , y ) d x d y$ 
B)  $\int _ { 9 } ^ { 4 5 } \int _ { 1 } ^ { y / 9 } f ( x , y ) d y d x$ 
C)  $\int _ { 9 } ^ { 4 5 } \int _ { 1 } ^ { y / 9 } f ( x , y ) d x d y$ 
D)  $\int _ { 9 } ^ { 45 } \int _ { 1 } ^ { 0 y } f ( x , y ) d x d y$

Question 17

Find the area of the part of the hyperbolic paraboloid $z = y ^ { 2 } - x ^ { 2 }$ that lies between the cylinders $x ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } + y ^ { 2 } = 16$ .

Accepted Answer

A)  $\frac { 65 \sqrt { 65 } - 5 \sqrt { 5 } } { 3 } \pi$   
B)  $\frac { \sqrt { 65 } - \sqrt { 5 } } { 6 } \pi$ 
C)  $\frac { 65 \sqrt { 65 } - 5 \sqrt { 5 } } { 6 } \pi$ 
D)  $\frac { \sqrt { 65 } - \sqrt { 5 } } { 3 } \pi$ 
E)  $\frac { 32 \sqrt { 65 } - 4 \sqrt { 5 } } { 3 } \pi$ 
A)  $\frac { 65 \sqrt { 65 } - 5 \sqrt { 5 } } { 3 } \pi$   
B)  $\frac { \sqrt { 65 } - \sqrt { 5 } } { 6 } \pi$ 
C)  $\frac { 65 \sqrt { 65 } - 5 \sqrt { 5 } } { 6 } \pi$ 
D)  $\frac { \sqrt { 65 } - \sqrt { 5 } } { 3 } \pi$ 
E)  $\frac { 32 \sqrt { 65 } - 4 \sqrt { 5 } } { 3 } \pi$

Question 18

Find the condition on the non-negative constants a and b for p(x, y)to be a joint density function, where $p ( x , y ) = \left\{ \begin{array} { c l } 
a x + b y , & 0 \leq x \leq 8,0 \leq y \leq 8 \
0 & \text { otherwise }
\end{array} \right.$

Accepted Answer

A)  $a + b = \frac { 1 } { 256 }$ 
B)  $a + b = \frac { 1 } { 32 }$ 
C)  $a + b < \frac { 1 } { 256 }$ 
D) a=b 
E)  $a + b = \frac { 1 } { 512 }$ 
A)  $a + b = \frac { 1 } { 256 }$ 
B)  $a + b = \frac { 1 } { 32 }$ 
C)  $a + b < \frac { 1 } { 256 }$ 
D) a=b 
E)  $a + b = \frac { 1 } { 512 }$

Question 19

Sketch the region of integration of the following integral and then convert the expression to polar co-ordinates (you do not have to evaluate it). $\int _ { 0 } ^ { \sqrt { 2 } } \int _ { x } ^ { \sqrt { 4 - x ^ { 2 } } } x ^ { 5 } y ^ { 5 } d y d x$

Accepted Answer

The area is shown be

Question 20

Calculate the following integral exactly.(Your answer should not be a decimal approximating the true answer, but should be exactly equal to the true answer.Your answer may contain e,  $\pi$, $\sqrt { 2 }$ , and so on.) $\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { 3 } \cos 4 y \sin ( 4 x + 5 ) d x d y$

Accepted Answer

The answer of Calculate the following integral exactly.(Your answer should...

Find a region R such that double integral $\int _ { R } \left( 2 - x ^ { 2 } - y ^ { 2 } \right) d A$ has the largest value.

Consider the change of variables x = s + 5t, y = s - 3t. Find the absolute value of the Jacobian $\left| \frac { \partial ( x , y ) } { \partial ( s , t ) } \right|$ .

Let W be the region between the cylinders $x ^ { 2 } + y ^ { 2 } = 9$ and $x ^ { 2 } + y ^ { 2 } = 36$ in the first octant and under the plane z = 1.Evaluate $\int _ { W } Z d V$

Set up but do not evaluate a (multiple)integral that gives the volume of the solid bounded above by the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 2$ and below by the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ .

Consider the integral $\int _ { - 6 } ^ { 6 } \int _ { 6 } ^ { \sqrt { 72 - x ^ { 2 } } } f ( x , y ) d y d x$ .Convert the integral to polar coordinates.

Evaluate the integral by interchanging the order of integration. $\int _ { - 1 } ^ { 0 } \int _ { - 4 x } ^ { 4 } e ^ { y ^ { 2 } } d y d x$ .

Rewrite the integral $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { \sqrt { 4 - y ^ { 2 } } } \int _ { \sqrt { x ^ { 2 } + y ^ { 2 } } } ^ { \sqrt { 8 - x ^ { 2 } - y ^ { 2 } } } \left( x ^ { 2 } + y ^ { 2 } \right) d z d x d y$ in spherical coordinates.You do not have to evaluate the integral.

If f and g are two continuous functions on a region R, then $\int _ { R } f \cdot g d A = \int _ { R } f d A \cdot \int _ { R } g d A$ .

For the following region, decide whether to integrate using polar or Cartesian coordinates.Write an iterated integral of an arbitrary function f(x, y)over the region.

Evaluate the integral $\int _ { x } x + 3 y d A$ , where R is the region in the first quadrant bounded by the y-axis, the line y = x and the circles $x ^ { 2 } + y ^ { 2 } = 25 , x ^ { 2 } + y ^ { 2 } = 16$

Convert the integral $\int _ { - 3 } ^ { 3 } \int _ { 0 } ^ { \sqrt { 9 - x ^ { 2 } } } e ^ { - \left( x ^ { 2 } + y ^ { 2 } \right) } d y d x$ to polar coordinates and hence evaluate it exactly.

Let W be the part of the solid sphere of radius 4, centered at the origin, that lies above the plane z = 2.Express $\int _ { W } z d V$ in (a)Cartesian (b)Cylindrical (c)Spherical coordinates.

Evaluate the integral $\int _ { 0 } ^ { 2 } \int _ { - \sqrt { 4 - x ^ { 2 } } } ^ { 0 } \int _ { - \sqrt { 4 - x ^ { 2 } - y ^ { 2 } } } ^ { \sqrt { 4 - x ^ { 2 } - y ^ { 2 } } } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } d z d y d x$ in spherical coordinates.

Let R₁ be the region 0 $\le$ x $\le$ 3, -2 $\le$ y $\le$ 4, and let R₂ be the region 3 $\le$ x $\le$ 5, -2 $\le$ y $\le$ 4.Suppose that the average value of f over R₁ is 6 and the average value over R₂is 7. Find the average value of f over R, $0 \leq x \leq 5 , - 2 \leq y \leq 2$ .

Consider the integral $\int _ { 1 } ^ { 5 } \int _ { 9 x } ^ { 45 } f ( x , y ) d y d x$ Rewrite the integral with the integration performed in the opposite order.

Find the area of the part of the hyperbolic paraboloid $z = y ^ { 2 } - x ^ { 2 }$ that lies between the cylinders $x ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } + y ^ { 2 } = 16$ .

Find the condition on the non-negative constants a and b for p(x, y)to be a joint density function, where $p ( x , y ) = \left\{ \begin{array} { c l } a x + b y , & 0 \leq x \leq 8,0 \leq y \leq 8 \\0 & \text { otherwise }\end{array} \right.$

Sketch the region of integration of the following integral and then convert the expression to polar co-ordinates (you do not have to evaluate it). $\int _ { 0 } ^ { \sqrt { 2 } } \int _ { x } ^ { \sqrt { 4 - x ^ { 2 } } } x ^ { 5 } y ^ { 5 } d y d x$

Calculate the following integral exactly.(Your answer should not be a decimal approximating the true answer, but should be exactly equal to the true answer.Your answer may contain e, $\pi$ , $\sqrt { 2 }$ , and so on.) $\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { 3 } \cos 4 y \sin ( 4 x + 5 ) d x d y$

A Library of Functions

Key Concept: the Derivative

Short-Cuts to Differentiation

Using the Derivative

Key Concept- the Definite Integral

Constructing Antiderivatives

Integration

Using the Definite Integral

Sequences and Series

Approximating Functions Using Series

Differential Equations

Functions of Several Variables

A Fundamental Tool- Vectors

Differentiating Functions of Several Variables

Optimization- Local and Global Extrema

Parameterization and Vector Fields

Line Integrals

Flux Integrals and Divergence

The Curl and Stokes Theorem

Parameters, Coordinates, Integrals

Filters

Exam 16: Integrating Functions of Several Variables

Find a region R such that double integral ∫R(2−x2−y2)dA\int _ { R } \left( 2 - x ^ { 2 } - y ^ { 2 } \right) d A∫R​(2−x2−y2)dA has the largest value.

Consider the change of variables x = s + 5t, y = s - 3t. Find the absolute value of the Jacobian ∣∂(x,y)∂(s,t)∣\left| \frac { \partial ( x , y ) } { \partial ( s , t ) } \right|​∂(s,t)∂(x,y)​​ .

Let W be the region between the cylinders x2+y2=9x ^ { 2 } + y ^ { 2 } = 9x2+y2=9 and x2+y2=36x ^ { 2 } + y ^ { 2 } = 36x2+y2=36 in the first octant and under the plane z = 1.Evaluate ∫WZdV\int _ { W } Z d V∫W​ZdV

Set up but do not evaluate a (multiple)integral that gives the volume of the solid bounded above by the sphere x2+y2+z2=2x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 2x2+y2+z2=2 and below by the paraboloid z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 .

Consider the integral ∫−66∫672−x2f(x,y)dydx\int _ { - 6 } ^ { 6 } \int _ { 6 } ^ { \sqrt { 72 - x ^ { 2 } } } f ( x , y ) d y d x∫−66​∫672−x2​​f(x,y)dydx .Convert the integral to polar coordinates.

Evaluate the integral by interchanging the order of integration. ∫−10∫−4x4ey2dydx\int _ { - 1 } ^ { 0 } \int _ { - 4 x } ^ { 4 } e ^ { y ^ { 2 } } d y d x∫−10​∫−4x4​ey2dydx .

If f and g are two continuous functions on a region R, then ∫Rf⋅gdA=∫RfdA⋅∫RgdA\int _ { R } f \cdot g d A = \int _ { R } f d A \cdot \int _ { R } g d A∫R​f⋅gdA=∫R​fdA⋅∫R​gdA .

For the following region, decide whether to integrate using polar or Cartesian coordinates.Write an iterated integral of an arbitrary function f(x, y)over the region.

Evaluate the integral ∫xx+3ydA\int _ { x } x + 3 y d A∫x​x+3ydA , where R is the region in the first quadrant bounded by the y-axis, the line y = x and the circles x2+y2=25,x2+y2=16x ^ { 2 } + y ^ { 2 } = 25 , x ^ { 2 } + y ^ { 2 } = 16x2+y2=25,x2+y2=16

Convert the integral ∫−33∫09−x2e−(x2+y2)dydx\int _ { - 3 } ^ { 3 } \int _ { 0 } ^ { \sqrt { 9 - x ^ { 2 } } } e ^ { - \left( x ^ { 2 } + y ^ { 2 } \right) } d y d x∫−33​∫09−x2​​e−(x2+y2)dydx to polar coordinates and hence evaluate it exactly.

Let W be the part of the solid sphere of radius 4, centered at the origin, that lies above the plane z = 2.Express ∫WzdV\int _ { W } z d V∫W​zdV in (a)Cartesian (b)Cylindrical (c)Spherical coordinates.

Consider the integral ∫15∫9x45f(x,y)dydx\int _ { 1 } ^ { 5 } \int _ { 9 x } ^ { 45 } f ( x , y ) d y d x∫15​∫9x45​f(x,y)dydx Rewrite the integral with the integration performed in the opposite order.

Find the area of the part of the hyperbolic paraboloid z=y2−x2z = y ^ { 2 } - x ^ { 2 }z=y2−x2 that lies between the cylinders x2+y2=1x ^ { 2 } + y ^ { 2 } = 1x2+y2=1 and x2+y2=16x ^ { 2 } + y ^ { 2 } = 16x2+y2=16 .

A Library of Functions

Key Concept: the Derivative

Short-Cuts to Differentiation

Using the Derivative

Key Concept- the Definite Integral

Constructing Antiderivatives

Integration

Using the Definite Integral

Sequences and Series

Approximating Functions Using Series

Differential Equations

Functions of Several Variables

A Fundamental Tool- Vectors

Differentiating Functions of Several Variables

Optimization- Local and Global Extrema

Parameterization and Vector Fields

Line Integrals

Flux Integrals and Divergence

The Curl and Stokes Theorem

Parameters, Coordinates, Integrals

Filters

Find a region R such that double integral $\int _ { R } \left( 2 - x ^ { 2 } - y ^ { 2 } \right) d A$ has the largest value.

Consider the change of variables x = s + 5t, y = s - 3t. Find the absolute value of the Jacobian $\left| \frac { \partial ( x , y ) } { \partial ( s , t ) } \right|$ .

Let W be the region between the cylinders $x ^ { 2 } + y ^ { 2 } = 9$ and $x ^ { 2 } + y ^ { 2 } = 36$ in the first octant and under the plane z = 1.Evaluate $\int _ { W } Z d V$

Set up but do not evaluate a (multiple)integral that gives the volume of the solid bounded above by the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 2$ and below by the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ .

Consider the integral $\int _ { - 6 } ^ { 6 } \int _ { 6 } ^ { \sqrt { 72 - x ^ { 2 } } } f ( x , y ) d y d x$ .Convert the integral to polar coordinates.

Evaluate the integral by interchanging the order of integration. $\int _ { - 1 } ^ { 0 } \int _ { - 4 x } ^ { 4 } e ^ { y ^ { 2 } } d y d x$ .

If f and g are two continuous functions on a region R, then $\int _ { R } f \cdot g d A = \int _ { R } f d A \cdot \int _ { R } g d A$ .

Evaluate the integral $\int _ { x } x + 3 y d A$ , where R is the region in the first quadrant bounded by the y-axis, the line y = x and the circles $x ^ { 2 } + y ^ { 2 } = 25 , x ^ { 2 } + y ^ { 2 } = 16$

Convert the integral $\int _ { - 3 } ^ { 3 } \int _ { 0 } ^ { \sqrt { 9 - x ^ { 2 } } } e ^ { - \left( x ^ { 2 } + y ^ { 2 } \right) } d y d x$ to polar coordinates and hence evaluate it exactly.

Let W be the part of the solid sphere of radius 4, centered at the origin, that lies above the plane z = 2.Express $\int _ { W } z d V$ in (a)Cartesian (b)Cylindrical (c)Spherical coordinates.

Consider the integral $\int _ { 1 } ^ { 5 } \int _ { 9 x } ^ { 45 } f ( x , y ) d y d x$ Rewrite the integral with the integration performed in the opposite order.

Find the area of the part of the hyperbolic paraboloid $z = y ^ { 2 } - x ^ { 2 }$ that lies between the cylinders $x ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } + y ^ { 2 } = 16$ .