Question 1

Find the Jacobian of the transformation $x = u - v ^ { 2 }$ , $y = u + v ^ { 2 }$ .

Accepted Answer

4v

Question 2

Find the Jacobian of the transformation $x = \rho \sin \phi \cos \theta$ , $y = \rho \sin \phi \sin \theta$ , $z = \rho \cos \phi$ .
A)$\theta$
B) $\beta ^ { 2 } \sin \phi$ 
C) $\rho \sin \phi$
D) $\rho ^ { 2 } \cos \phi$ 
E)2$\theta$
F) $\rho ^ { 2 } \sin \theta$  
G) $\rho \sin \theta$ 
H) $\rho \cos \phi$

Accepted Answer

B

Question 3

Find the Jacobian of the transformation $x = s e ^ { t }$ , $y = s e ^ { - t }$ .

Accepted Answer

-2s

Question 4

Evaluate the integral $\iint _ { R } \sqrt { x ^ { 2 } + 9 y ^ { 2 } } d A$ , where R is the region enclosed by the ellipse $\frac { x ^ { 2 } } { 9 } + y ^ { 2 } = 1$ .
A)24$\pi$
B)$\pi$
C)12$\pi$
D)3$\pi$
E)10$\pi$
F)2$\pi$
G)6$\pi$
H)4$\pi$

Accepted Answer

The answer of Evaluate the integral \(\iint _ { R...

Question 5

Find the Jacobian of the transformation $x = u ^ { 2 }$ , $y = v ^ { 3 }$ .
A) $u v ^ { 2 }$

B) $2 u v ^ { 2 }$ 
C) $3 u v ^ { 2 }$
D)$6 u v ^ { 2 }$ 
E) $- u v ^ { 2 }$ 
F) $- 2 u v ^ { 2 }$

G) $- 3 u v ^ { 2 }$ 
 
H) $- 6 u v ^ { 2 }$

Accepted Answer

The Jacobian is calculated as the determinant of the matrix of partial derivatives: $\frac{\partial (x, y)}{\partial (u, v)} = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix} = \begin{vmatrix} 2u & 0 \ 0 & 3v^2 \end{vmatrix} = 6uv^2$.

Question 6

Find the Jacobian of the transformation x = 3u + v, y = u - 2w, z = v + w.
A)1
B)6
C)-5
D)4
E)-1
F)-6
G)5
H)-4

Accepted Answer

The answer of Find the Jacobian of the transformation x...

Question 7

Find the area of the region whose image under the transformation x = u + v, y = v - 2u is $D = \left\{ ( x , y ) \mid - 1 \leq x \leq 1,0 \leq y \leq 1 - x ^ { 2 } \right\}$ .
A) $\frac { 1 } { 9 }$

B) $\frac { 2 } { 9 }$ 
C) $\frac { 1 } { 3 }$ 
D) $\frac { 4 } { 9 }$ 
E) $\frac { 5 } { 9 }$ 
F) $\frac { 2 } { 3 }$

G) $\frac { 7 } { 9 }$

H) $\frac { 8 } { 9 }$

Accepted Answer

The answer of Find the area of the region whose...

Question 8

Evaluate the integral $\iint _ { R } \frac { x + y } { x - y } d A$ , where R is the triangular region with vertices (1, 0), (0, -1), and (0, 0).
A)0
B) $\frac { 1 } { 4 }$
C) $\frac { 1 } { 2 }$ 
D) $\frac { 3 } { 4 }$ 
E)1
F) $\frac { 3 } { 2 }$ 
G)2
H)-2

Accepted Answer

The integral simplifies due to the symmetry of the function over the triangular region. The region is symmetric about the line y = -x, and the integrand function $\frac{x+y}{x-y}$ changes sign when x and y are swapped, making the integral over the symmetric region equal to zero.

Question 9

Find the Jacobian of the transformation $x = u ^ { 2 }$ , $y = v ^ { 3 }$ , when $u = \frac { 1 } { 2 }$ and v = 1.
A)1
B)2
C)3
D)6
E)-1
F)-2
G)-3
H)-6

Accepted Answer

The answer of Find the Jacobian of the transformation \(x...

Question 10

Find the Jacobian of the transformation x = u + v, y = 2u - v.
A)1
B)2
C)3
D)4
E)-1
F)-2
G)-3
H)-4

Accepted Answer

The answer of Find the Jacobian of the transformation x...

Question 11

Use the change of variables u = 2x - y, v = x + y to evaluate $\iint _ { R } ( 6 x - 3 y ) d A$ where R is the region bounded by 2x - y = 1, 2x - y = 3, x + y = 1, and x + y = 2.

Accepted Answer

The answer of Use the change of variables u =...

Question 12

Find the Jacobian of the transformation x = u, y = 2v, z = 3w.
A)1
B)2
C)3
D)6
E)-1
F)-2
G)-3
H)-6

Accepted Answer

The Jacobian of the transformation is the determinant of the matrix of partial derivatives. We have:

| 1  0  0 |
| 0  2  0 |
| 0  0  3 |

The determinant of this matrix is 1 x 2 x 3 = 6, so the answer is D, 6.

Question 13

Use the change of variables $u = x y$ , $v = x y ^ { 2 }$ to evaluate $\iint _ { R } y ^ { 2 } d A$ , where R is the region bounded by the curves xy = 1, xy = 2, $x y ^ { 2 } = 1$ , and $x y ^ { 2 } = 2$ .

Accepted Answer

The answer of Use the change of variables \(u =...

Question 14

Under the transformation x = u + v, y = v - 2u, the image of the circle $x ^ { 2 } + y ^ { 2 } \leq 1$ is an ellipse. What is the area of that ellipse?
A) $\frac { \pi } { 4 }$ 
 
B) $\frac { \pi } { 3 }$ 
C) $\frac { \pi } { 2 }$ 
D) $\frac { 2 \pi } { 3 }$ 
E) $\pi$
F) $\frac { 3 \pi } { 2 }$

G) $2 \pi$

H) $3 \pi$

Accepted Answer

The answer of Under the transformation x = u +...

Question 15

Use the change of variables x = 2u + 3v, y = 3u - 2v to evaluate $\iint _ { R } ( x + y ) d A$ , where R is the square with vertices (0, 0), (2, 3), (5, 1), and (3, -2).

Accepted Answer

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

Question 16

Find the Jacobian of the transformation x = u sin v, y = u cos v when u = 3 and v = 5.
A)3
B)5
C)7.5
D)15
E)-3
F)-5
G)-7.5
H)-15

Accepted Answer

The answer of Find the Jacobian of the transformation x...

Question 17

Find the Jacobian of the transformation $x = r \cos \theta$ , $y = r \sin \theta$ , z = z.
A)$\theta$
B)z
C)r
D)1
E)2$\theta$
F)2z
G)2r
H)2

Accepted Answer

The Jacobian of the transformation $x = r \cos \theta$, $y = r \sin \theta$, $z = z$ is calculated by taking the determinant of the matrix formed by the partial derivatives of $x$, $y$, and $z$ with respect to $r$, $\theta$, and $z$, respectively. The partial derivatives are:$$\frac{\partial x}{\partial r} = \cos \theta, \quad \frac{\partial x}{\partial \theta} = -r \sin \theta, \quad \frac{\partial x}{\partial z} = 0$$$$\frac{\partial y}{\partial r} = \sin \theta, \quad \frac{\partial y}{\partial \theta} = r \cos \theta, \quad \frac{\partial y}{\partial z} = 0$$$$\frac{\partial z}{\partial r} = 0, \quad \frac{\partial z}{\partial \theta} = 0, \quad \frac{\partial z}{\partial z} = 1$$The Jacobian matrix is:$$\begin{vmatrix}\cos \theta & -r \sin \theta & 0 \\sin \theta & r \cos \theta & 0 \0 & 0 & 1\end{vmatrix}$$The determinant of this matrix is $1 \times (\cos \theta \cdot r \cos \theta + r \sin^2 \theta) = r(\cos^2 \theta + \sin^2 \theta) = r$, since $\cos^2 \theta + \sin^2 \theta = 1$. Therefore, the Jacobian of the transformation is $r$.

Question 18

Find the Jacobian of the transformation x = 2u, $y = 3 v ^ { 2 }$ , $z = 4 w ^ { 3 }$ .

Accepted Answer

The answer of Find the Jacobian of the transformation x...

Question 19

Find the Jacobian of the transformation $x = u \sin v$ , $y = u \cos v$ .
A) $- u$ 
 
B)
$- u ^ { 2 }$ 
C)u
D) $u ^ { 2 }$
E) $- u ^ { 2 } \sin v \cos v$
F) $u ^ { 2 } \sin v \cos v$ 
G) $- 2 u ^ { 2 } \sin v \cos v$  
H) $2 u ^ { 2 } \sin v \cos v$

Accepted Answer

The Jacobian of the transformation given by $x = u \sin v$ and $y = u \cos v$ is calculated as the determinant of the matrix formed by the partial derivatives of $x$ and $y$ with respect to $u$ and $v$. This gives:$$\begin{vmatrix}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}\end{vmatrix}=\begin{vmatrix}\sin v & u \cos v \\cos v & -u \sin v\end{vmatrix}= (-u \sin^2 v) - (u \cos^2 v) = -u (\sin^2 v + \cos^2 v) = -u$$Hence, the correct answer is $-u$.

Question 20

Use the change of variables $x = \sqrt { 2 } u - \sqrt { \frac { 2 } { 3 } } v$ , $y = \sqrt { 2 } u + \sqrt { \frac { 2 } { 3 } } v$ to evaluate $\iint _ { R } \left( x ^ { 2 } - x y + y ^ { 2 } \right) d A$ , where R is the region bounded by the ellipse $x ^ { 2 } - x y + y ^ { 2 } = 2$ .

Accepted Answer

The answer of Use the change of variables \(x =...

Quiz 12: Multiple Integrals

Find the Jacobian of the transformation $x = u - v ^ { 2 }$ , $y = u + v ^ { 2 }$ .

Find the Jacobian of the transformation $x = \rho \sin \phi \cos \theta$ , $y = \rho \sin \phi \sin \theta$ , $z = \rho \cos \phi$ .

Find the Jacobian of the transformation $x = s e ^ { t }$ , $y = s e ^ { - t }$ .

Evaluate the integral $\iint _ { R } \sqrt { x ^ { 2 } + 9 y ^ { 2 } } d A$ , where R is the region enclosed by the ellipse $\frac { x ^ { 2 } } { 9 } + y ^ { 2 } = 1$ .

Find the Jacobian of the transformation $x = u ^ { 2 }$ , $y = v ^ { 3 }$ .

Find the Jacobian of the transformation x = 3u + v, y = u - 2w, z = v + w.

Find the area of the region whose image under the transformation x = u + v, y = v - 2u is $D = \left\{ ( x , y ) \mid - 1 \leq x \leq 1,0 \leq y \leq 1 - x ^ { 2 } \right\}$ .

Evaluate the integral $\iint _ { R } \frac { x + y } { x - y } d A$ , where R is the triangular region with vertices (1, 0) , (0, -1) , and (0, 0) .

Find the Jacobian of the transformation $x = u ^ { 2 }$ , $y = v ^ { 3 }$ , when $u = \frac { 1 } { 2 }$ and v = 1.

Find the Jacobian of the transformation x = u + v, y = 2u - v.

Use the change of variables u = 2x - y, v = x + y to evaluate $\iint _ { R } ( 6 x - 3 y ) d A$ where R is the region bounded by 2x - y = 1, 2x - y = 3, x + y = 1, and x + y = 2.

Find the Jacobian of the transformation x = u, y = 2v, z = 3w.

Use the change of variables $u = x y$ , $v = x y ^ { 2 }$ to evaluate $\iint _ { R } y ^ { 2 } d A$ , where R is the region bounded by the curves xy = 1, xy = 2, $x y ^ { 2 } = 1$ , and $x y ^ { 2 } = 2$ .

Under the transformation x = u + v, y = v - 2u, the image of the circle $x ^ { 2 } + y ^ { 2 } \leq 1$ is an ellipse. What is the area of that ellipse?

Use the change of variables x = 2u + 3v, y = 3u - 2v to evaluate $\iint _ { R } ( x + y ) d A$ , where R is the square with vertices (0, 0), (2, 3), (5, 1), and (3, -2).

Find the Jacobian of the transformation x = u sin v, y = u cos v when u = 3 and v = 5.

Find the Jacobian of the transformation $x = r \cos \theta$ , $y = r \sin \theta$ , z = z.

Find the Jacobian of the transformation x = 2u, $y = 3 v ^ { 2 }$ , $z = 4 w ^ { 3 }$ .

Find the Jacobian of the transformation $x = u \sin v$ , $y = u \cos v$ .

Use the change of variables $x = \sqrt { 2 } u - \sqrt { \frac { 2 } { 3 } } v$ , $y = \sqrt { 2 } u + \sqrt { \frac { 2 } { 3 } } v$ to evaluate $\iint _ { R } \left( x ^ { 2 } - x y + y ^ { 2 } \right) d A$ , where R is the region bounded by the ellipse $x ^ { 2 } - x y + y ^ { 2 } = 2$ .

Quiz 12: Multiple Integrals

Find the Jacobian of the transformation x=u−v2x = u - v ^ { 2 }x=u−v2 , y=u+v2y = u + v ^ { 2 }y=u+v2 .

Find the Jacobian of the transformation x=ρsin⁡ϕcos⁡θx = \rho \sin \phi \cos \thetax=ρsinϕcosθ , y=ρsin⁡ϕsin⁡θy = \rho \sin \phi \sin \thetay=ρsinϕsinθ , z=ρcos⁡ϕz = \rho \cos \phiz=ρcosϕ .

Find the Jacobian of the transformation x=setx = s e ^ { t }x=set , y=se−ty = s e ^ { - t }y=se−t .

Evaluate the integral ∬Rx2+9y2dA\iint _ { R } \sqrt { x ^ { 2 } + 9 y ^ { 2 } } d A∬R​x2+9y2​dA , where R is the region enclosed by the ellipse x29+y2=1\frac { x ^ { 2 } } { 9 } + y ^ { 2 } = 19x2​+y2=1 .

Find the Jacobian of the transformation x=u2x = u ^ { 2 }x=u2 , y=v3y = v ^ { 3 }y=v3 .

Find the Jacobian of the transformation x = 3u + v, y = u - 2w, z = v + w.

Find the area of the region whose image under the transformation x = u + v, y = v - 2u is D={(x,y) ∣−1≤x≤1,0≤y≤1−x2}D = \left\{ ( x , y ) \mid - 1 \leq x \leq 1,0 \leq y \leq 1 - x ^ { 2 } \right\}D={(x,y) ∣−1≤x≤1,0≤y≤1−x2} .

Evaluate the integral ∬Rx+yx−ydA\iint _ { R } \frac { x + y } { x - y } d A∬R​x−yx+y​dA , where R is the triangular region with vertices (1, 0) , (0, -1) , and (0, 0) .

Find the Jacobian of the transformation x=u2x = u ^ { 2 }x=u2 , y=v3y = v ^ { 3 }y=v3 , when u=12u = \frac { 1 } { 2 }u=21​ and v = 1.

Find the Jacobian of the transformation x = u + v, y = 2u - v.

Use the change of variables u = 2x - y, v = x + y to evaluate ∬R(6x−3y)dA\iint _ { R } ( 6 x - 3 y ) d A∬R​(6x−3y)dA where R is the region bounded by 2x - y = 1, 2x - y = 3, x + y = 1, and x + y = 2.

Find the Jacobian of the transformation x = u, y = 2v, z = 3w.

Use the change of variables u=xyu = x yu=xy , v=xy2v = x y ^ { 2 }v=xy2 to evaluate ∬Ry2dA\iint _ { R } y ^ { 2 } d A∬R​y2dA , where R is the region bounded by the curves xy = 1, xy = 2, xy2=1x y ^ { 2 } = 1xy2=1 , and xy2=2x y ^ { 2 } = 2xy2=2 .

Under the transformation x = u + v, y = v - 2u, the image of the circle x2+y2≤1x ^ { 2 } + y ^ { 2 } \leq 1x2+y2≤1 is an ellipse. What is the area of that ellipse?

Use the change of variables x = 2u + 3v, y = 3u - 2v to evaluate ∬R(x+y)dA\iint _ { R } ( x + y ) d A∬R​(x+y)dA , where R is the square with vertices (0, 0), (2, 3), (5, 1), and (3, -2).

Find the Jacobian of the transformation x = u sin v, y = u cos v when u = 3 and v = 5.

Find the Jacobian of the transformation x=rcos⁡θx = r \cos \thetax=rcosθ , y=rsin⁡θy = r \sin \thetay=rsinθ , z = z.

Find the Jacobian of the transformation x = 2u, y=3v2y = 3 v ^ { 2 }y=3v2 , z=4w3z = 4 w ^ { 3 }z=4w3 .

Find the Jacobian of the transformation x=usin⁡vx = u \sin vx=usinv , y=ucos⁡vy = u \cos vy=ucosv .

Find the Jacobian of the transformation $x = u - v ^ { 2 }$ , $y = u + v ^ { 2 }$ .

Find the Jacobian of the transformation $x = \rho \sin \phi \cos \theta$ , $y = \rho \sin \phi \sin \theta$ , $z = \rho \cos \phi$ .

Find the Jacobian of the transformation $x = s e ^ { t }$ , $y = s e ^ { - t }$ .

Evaluate the integral $\iint _ { R } \sqrt { x ^ { 2 } + 9 y ^ { 2 } } d A$ , where R is the region enclosed by the ellipse $\frac { x ^ { 2 } } { 9 } + y ^ { 2 } = 1$ .

Find the Jacobian of the transformation $x = u ^ { 2 }$ , $y = v ^ { 3 }$ .

Find the area of the region whose image under the transformation x = u + v, y = v - 2u is $D = \left\{ ( x , y ) \mid - 1 \leq x \leq 1,0 \leq y \leq 1 - x ^ { 2 } \right\}$ .

Evaluate the integral $\iint _ { R } \frac { x + y } { x - y } d A$ , where R is the triangular region with vertices (1, 0) , (0, -1) , and (0, 0) .

Find the Jacobian of the transformation $x = u ^ { 2 }$ , $y = v ^ { 3 }$ , when $u = \frac { 1 } { 2 }$ and v = 1.

Use the change of variables u = 2x - y, v = x + y to evaluate $\iint _ { R } ( 6 x - 3 y ) d A$ where R is the region bounded by 2x - y = 1, 2x - y = 3, x + y = 1, and x + y = 2.

Use the change of variables $u = x y$ , $v = x y ^ { 2 }$ to evaluate $\iint _ { R } y ^ { 2 } d A$ , where R is the region bounded by the curves xy = 1, xy = 2, $x y ^ { 2 } = 1$ , and $x y ^ { 2 } = 2$ .

Under the transformation x = u + v, y = v - 2u, the image of the circle $x ^ { 2 } + y ^ { 2 } \leq 1$ is an ellipse. What is the area of that ellipse?

Use the change of variables x = 2u + 3v, y = 3u - 2v to evaluate $\iint _ { R } ( x + y ) d A$ , where R is the square with vertices (0, 0), (2, 3), (5, 1), and (3, -2).

Find the Jacobian of the transformation $x = r \cos \theta$ , $y = r \sin \theta$ , z = z.

Find the Jacobian of the transformation x = 2u, $y = 3 v ^ { 2 }$ , $z = 4 w ^ { 3 }$ .

Find the Jacobian of the transformation $x = u \sin v$ , $y = u \cos v$ .