Lesson test: Change of variables

Change of variables

1.[1p]
Which area element converts a double integral to polar coordinates?

$d A = r d r d θ$

Correct

$d A = d r d θ$

The answer is: $dA = r\,dr\,d\theta$

$d A = r^{2} d r d θ$

The answer is: $dA = r\,dr\,d\theta$

$d A = \frac{1}{r} d r d θ$

The answer is: $dA = r\,dr\,d\theta$

2.[2p]
A thin disc of radius $3$ has surface density $ρ = \sqrt{x^{2} + y^{2}}$ . Find its mass, to two decimal places.

CorrectNot quite: 56.55

3.[2p]
Evaluate $\iint_{A} (x^{2} + y^{2}) d A$ over the annulus $4 \leq x^{2} + y^{2} \leq 9$ , to two decimal places.

CorrectNot quite: 102.10

4.[2p]
Evaluate $\iint_{Q} y d A$ , where $Q$ is the quarter of the disc $x^{2} + y^{2} \leq 9$ lying in the first quadrant.

CorrectNot quite: 9

5.[2p]
Given that $\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt{π}$ , evaluate $\int_{- \infty}^{\infty} e^{- 4 x^{2}} d x$ , to four decimal places.

CorrectNot quite: 0.8862

6.[2p]
Find the Jacobian $\frac{\partial x}{\partial u} \frac{\partial y}{\partial v} - \frac{\partial x}{\partial v} \frac{\partial y}{\partial u}$ of the substitution $x = u^{2} - v^{2}$ , $y = 2 u v$ at the point $(u, v) = (1, 2)$ .

CorrectNot quite: 20

7.[2p]
Find the area enclosed by the ellipse $\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$ , to two decimal places.

CorrectNot quite: 37.70

8.[3p]
Let $R$ be the region bounded by the lines $x + y = 0$ , $x + y = 4$ , $x - y = 0$ and $x - y = 2$ . Evaluate $\iint_{R} (x + y) d A$ .

CorrectNot quite: 8

9.[3p]
Which of these statements about the Jacobian $J$ of a substitution $x = x (u, v)$ , $y = y (u, v)$ are true?

Select all that apply

Near each point, the substitution multiplies small areas by $| J |$

Correct

For polar coordinates, $J = r$

Correct

For a linear substitution, $J$ is the same at every point

Correct

The change of variables formula uses $J$ without the absolute value, as one variable substitution uses $g^{'} (u)$

The answer is: Near each point, the substitution multiplies small areas by $|J|$, For polar coordinates, $J = r$, For a linear substitution, $J$ is the same at every point

$J = 1$ for every substitution that is one to one

The answer is: Near each point, the substitution multiplies small areas by $|J|$, For polar coordinates, $J = r$, For a linear substitution, $J$ is the same at every point

1.[1p] Which area element converts a double integral to polar coordinates?

2.[2p] A thin disc of radius 3 has surface density ρ=x2+y2. Find its mass, to two decimal places.

3.[2p] Evaluate ∬A(x2+y2)dA over the annulus 4≤x2+y2≤9, to two decimal places.

4.[2p] Evaluate ∬QydA, where Q is the quarter of the disc x2+y2≤9 lying in the first quadrant.

5.[2p] Given that ∫-∞∞e-x2dx=π, evaluate ∫-∞∞e-4x2dx, to four decimal places.

6.[2p] Find the Jacobian ∂x∂u∂y∂v-∂x∂v∂y∂u of the substitution x=u2-v2, y=2uv at the point (u,v)=(1,2).

7.[2p] Find the area enclosed by the ellipse x216+y29=1, to two decimal places.

8.[3p] Let R be the region bounded by the lines x+y=0, x+y=4, x-y=0 and x-y=2. Evaluate ∬R(x+y)dA.

9.[3p] Which of these statements about the Jacobian J of a substitution x=x(u,v), y=y(u,v) are true?

1.[1p]
Which area element converts a double integral to polar coordinates?

2.[2p]
A thin disc of radius $3$ has surface density $ρ = \sqrt{x^{2} + y^{2}}$ . Find its mass, to two decimal places.

3.[2p]
Evaluate $\iint_{A} (x^{2} + y^{2}) d A$ over the annulus $4 \leq x^{2} + y^{2} \leq 9$ , to two decimal places.

4.[2p]
Evaluate $\iint_{Q} y d A$ , where $Q$ is the quarter of the disc $x^{2} + y^{2} \leq 9$ lying in the first quadrant.

5.[2p]
Given that $\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt{π}$ , evaluate $\int_{- \infty}^{\infty} e^{- 4 x^{2}} d x$ , to four decimal places.

6.[2p]
Find the Jacobian $\frac{\partial x}{\partial u} \frac{\partial y}{\partial v} - \frac{\partial x}{\partial v} \frac{\partial y}{\partial u}$ of the substitution $x = u^{2} - v^{2}$ , $y = 2 u v$ at the point $(u, v) = (1, 2)$ .

7.[2p]
Find the area enclosed by the ellipse $\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$ , to two decimal places.

8.[3p]
Let $R$ be the region bounded by the lines $x + y = 0$ , $x + y = 4$ , $x - y = 0$ and $x - y = 2$ . Evaluate $\iint_{R} (x + y) d A$ .

9.[3p]
Which of these statements about the Jacobian $J$ of a substitution $x = x (u, v)$ , $y = y (u, v)$ are true?