Question 1

Which property of double integrals allows you to split the integral over a region $$R$$ into the sum of integrals over two subregions $$S$$ and $$T$$?

Accepted Answer

Additivity over regions: $$\iint_R f(x, y)\, dA = \iint_S f(x, y)\, dA + \iint_T f(x, y)\, dA$$

Question 2

Given $$f(x, y) = x$$ and $$R = [0, 2] 	imes [0, 1]$$, what is $$\iint_R f(x, y)\, dA$$?

Accepted Answer

$$2$$

Question 3

Which theorem allows you to compute a double integral as an iterated integral in either order for a continuous function on a rectangle?

Accepted Answer

Fubini's Theorem

Question 4

If $$f(x, y) \geq g(x, y)$$ for all $$(x, y) in$$ $$R$$, which of the following is true?

Accepted Answer

$$\iint_R f(x, y)\, dA \geq \iint_R g(x, y)\, dA$$

Question 5

Compute $$\int_{-4}^{4} 5x^3$\, dx.$$

Accepted Answer

$$0$$

Question 6

For $$f(x, y) = x^3$ and R$$ = [-4, 4] 	imes [0, 5]$$, what is $$\iint_R f(x, y)\, dA$$?

Accepted Answer

$$0$$

Question 7

Which property of double integrals states that $$\iint_R k f(x, y)\, dA = k \iint_R f(x, y)\, dA$$ for constant $$k$$?

Accepted Answer

Homogeneity (scalar multiplication)

Question 8

Suppose $$f(x, y) is $$continuous on $$R = [a, b] 	imes [c, d]. $$According to Fubini's Theorem, which of the following is always true?

Accepted Answer

$$\iint_R f(x, y)\, dA = \int_c^d \int_a^b f(x, y)\, dx\, dy$$

Question 9

Compute $$\$$int_0$$^1 e$$^{x-1}$$\, dx.$$

Accepted Answer

$$1 - e$$^{-1}$$

Question 10

A double integral $$\iint_R f(x, y)\, dA$$ gives which physical quantity for $$f(x, y)$$ defined on a region $$R$$?

Accepted Answer

The signed volume between the $$xy-$$plane and the graph of $$f(x, y)$$

Question 11

For $$f(x, y) = x e$$^{xy}$$ and R$$ = [0, 1] 	imes [0, 1]$$, what is $$\iint_R f(x, y)\, dA$$?

Accepted Answer

$$\frac{e - 1}{2}$$

Question 12

If you fix $$x = x_0$ and integrate f$$($$x_0$$, y)$$ from y = c$ to y = d$, what does this represent?

Accepted Answer

The cross-sectional (signed) area at $$x_0$$

Question 13

Which of the following is NOT a property of double integrals?

Accepted Answer

Differentiability: $$\iint_R f(x, y)\, dA = \frac{d}{dx} \iint_R f(x, y)\, dA$$

Question 14

A rectangular region $$R = [a, b] 	imes [c, d] is $$partitioned into two regions $$S$$ and $$T. If$$ $$\iint_S f(x, y)\, dA = 3$$ and $$\iint_T f(x, y)\, dA = 5$$, what is $$\iint_R f(x, y)\, dA$$?

Accepted Answer

$$8$$

Question 15

Which of the following best describes the process of approximating a double integral by summing thin slices?

Accepted Answer

Riemann sum approximation: $$\iint_R f(x, y)\, dA \approx \sum A(x^*) \Delta x$$

Question 16

If $$f(x, y) is $$continuous on $$R = [a, b] 	imes [c, d]$$, which order of integration is valid according to Fubini's Theorem?

Accepted Answer

$$\int_a^b \int_c^d f(x, y)\, dy\, dx$$

Question 17

Which scenario would require careful consideration of the order of integration when using Fubini's Theorem?

Accepted Answer

When the limits of integration are not constant and depend on the other variable

Question 18

Suppose you are modeling the heat distribution on a metal plate using $$f(x, y)$$, and you want to find the total heat content. Which mathematical operation would you use?

Accepted Answer

Double integral: $$\iint_R f(x, y)\, dA$$

Iterated Integrals and Properties of Double Integrals

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