Lesson test: Double integrals

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1.[1p]
Estimate $\iint_{R} x y d A$ over $R = [0, 2] \times [0, 2]$ with a $2 \times 2$ grid of unit squares, sampling each square at its upper right corner.

CorrectNot quite: 9

2.[2p]
Evaluate $\iint_{R} 6 x y^{2} d A$ over $R = [0, 2] \times [1, 3]$ .

CorrectNot quite: 104

3.[2p]
Evaluate $\iint_{R} x e^{x y} d A$ over $R = [0, 2] \times [0, 1]$ , choosing the order of integration that avoids integration by parts. Give the answer to three decimal places.

CorrectNot quite: 4.389

4.[2p]
Evaluate $\iint_{D} 6 x y d A$ , where $D$ is the region enclosed by the curves $y = x^{2}$ and $y = x$ .

CorrectNot quite: 0.25

5.[2p]
Which iterated integral equals $\int_{0}^{4} \int_{\sqrt{x}}^{2} f (x, y) d y d x$ with the order of integration reversed?

$\int_{0}^{2} \int_{0}^{y^{2}} f (x, y) d x d y$

Correct

$\int_{0}^{2} \int_{y^{2}}^{4} f (x, y) d x d y$

The answer is: $\int_0^2 \int_0^{y^2} f(x, y)\,dx\,dy$

$\int_{0}^{4} \int_{0}^{y^{2}} f (x, y) d x d y$

The answer is: $\int_0^2 \int_0^{y^2} f(x, y)\,dx\,dy$

$\int_{0}^{2} \int_{0}^{\sqrt{y}} f (x, y) d x d y$

The answer is: $\int_0^2 \int_0^{y^2} f(x, y)\,dx\,dy$

6.[3p]
Evaluate $\int_{0}^{2} \int_{x}^{2} e^{y^{2}} d y d x$ by reversing the order of integration. Give the answer to two decimal places.

CorrectNot quite: 26.80

7.[2p]
Find the average value of $f (x, y) = x^{2} + y^{2}$ over the rectangle $[0, 1] \times [0, 2]$ . Give the answer to three decimal places.

CorrectNot quite: 1.667

8.[3p]
A thin plate occupies the triangle with corners $(0, 0)$ , $(1, 0)$ and $(0, 1)$ , lengths in metres, and has density $ρ = 1 + x + y$ kilograms per square metre. What is its mass in kilograms, to three decimal places?

CorrectNot quite: 0.833

9.[3p]
Which of these statements are true?

Select all that apply

If $f$ is continuous on a rectangle, the two iterated integrals of $f$ over it are equal

Correct

$\iint_{D} 1 d A$ is the area of $D$

Correct

If $f (x, y) = g (x) h (y)$ on $[a, b] \times [c, d]$ , the double integral is $\int_{a}^{b} g (x) d x$ times $\int_{c}^{d} h (y) d y$

Correct

Whenever both iterated integrals of a function over the unit square exist, they are equal

The answer is: If $f$ is continuous on a rectangle, the two iterated integrals of $f$ over it are equal, $\iint_D 1\,dA$ is the area of $D$, If $f(x, y) = g(x)\,h(y)$ on $[a, b] \times [c, d]$, the double integral is $\int_a^b g(x)\,dx$ times $\int_c^d h(y)\,dy$

In an iterated integral, the outer limits may depend on the inner variable

$\int_{0}^{1} \int_{y}^{1} e^{x^{2}} d x d y$ has no exact value, because $e^{x^{2}}$ has no elementary antiderivative

1.[1p] Estimate ∬RxydA over R=[0,2]×[0,2] with a 2×2 grid of unit squares, sampling each square at its upper right corner.

2.[2p] Evaluate ∬R6xy2dA over R=[0,2]×[1,3].

3.[2p] Evaluate ∬RxexydA over R=[0,2]×[0,1], choosing the order of integration that avoids integration by parts. Give the answer to three decimal places.

4.[2p] Evaluate ∬D6xydA, where D is the region enclosed by the curves y=x2 and y=x.

5.[2p] Which iterated integral equals ∫04∫x2f(x,y)dydx with the order of integration reversed?

6.[3p] Evaluate ∫02∫x2ey2dydx by reversing the order of integration. Give the answer to two decimal places.

7.[2p] Find the average value of f(x,y)=x2+y2 over the rectangle [0,1]×[0,2]. Give the answer to three decimal places.

8.[3p] A thin plate occupies the triangle with corners (0,0), (1,0) and (0,1), lengths in metres, and has density ρ=1+x+y kilograms per square metre. What is its mass in kilograms, to three decimal places?

9.[3p] Which of these statements are true?

1.[1p]
Estimate $\iint_{R} x y d A$ over $R = [0, 2] \times [0, 2]$ with a $2 \times 2$ grid of unit squares, sampling each square at its upper right corner.

2.[2p]
Evaluate $\iint_{R} 6 x y^{2} d A$ over $R = [0, 2] \times [1, 3]$ .

3.[2p]
Evaluate $\iint_{R} x e^{x y} d A$ over $R = [0, 2] \times [0, 1]$ , choosing the order of integration that avoids integration by parts. Give the answer to three decimal places.

4.[2p]
Evaluate $\iint_{D} 6 x y d A$ , where $D$ is the region enclosed by the curves $y = x^{2}$ and $y = x$ .

5.[2p]
Which iterated integral equals $\int_{0}^{4} \int_{\sqrt{x}}^{2} f (x, y) d y d x$ with the order of integration reversed?

6.[3p]
Evaluate $\int_{0}^{2} \int_{x}^{2} e^{y^{2}} d y d x$ by reversing the order of integration. Give the answer to two decimal places.

7.[2p]
Find the average value of $f (x, y) = x^{2} + y^{2}$ over the rectangle $[0, 1] \times [0, 2]$ . Give the answer to three decimal places.

8.[3p]
A thin plate occupies the triangle with corners $(0, 0)$ , $(1, 0)$ and $(0, 1)$ , lengths in metres, and has density $ρ = 1 + x + y$ kilograms per square metre. What is its mass in kilograms, to three decimal places?

9.[3p]
Which of these statements are true?