Q: Which of the following statements about Fubini's Theorem is correct?

If $$f(x, y) is $$continuous on $$R = [a, b] \times [c, d]$$, then $$\iint_R f(x, y)\, dA = \int_a^b \left( \int_c^d f(x, y)\, dy \right) dx = \int_c^d \left( \int_a^b f(x, y)\, dx \right) dy$$

Question 1

Which of the following best describes the process of approximating the volume under a surface $$z = f(x, y)$$ over a rectangular region $$R = [a, b] 	imes [c, d]$$ using a Riemann sum?

Accepted Answer

Divide $$R$$ into small rectangles, pick a sample point in each, compute $$f(x_i, y_j) at $$each point, multiply by the area of each rectangle, and sum all the products.

Question 2

Given $$f(x, y) = 16 - x^2$ - y^2$ over R$$ = [0, 2] 	imes [0, 3]$$, what is the value of the double integral $$\iint_R f(x, y)\, dA$$?

Accepted Answer

$$70$$

Question 3

Which of the following is the correct formula for the average value of a continuous function $$f(x, y)$$ over a rectangle $$R = [a, b] 	imes [c, d]$$?

Accepted Answer

$$\displaystyle \frac{1}{(b-a)(d-c)} \iint_R f(x, y)\, dA$$

Question 4

Suppose you want to estimate the volume under $$z = 16 - x - y$$ over $$R = [0, 2] 	imes [0, 3]$$ using a Riemann sum with $$2$$ intervals on the $$x-$$axis and $$3$$ intervals on the $$y-$$axis, using the bottom-left corner as sample points. What is the area of each rectangle?

Accepted Answer

$$1$$

Question 5

Evaluate $$\iint_R (2x - y)\, dA$$ where $$R = [0, 2] 	imes [0, 3].$$

Accepted Answer

$$6$$

Question 6

Which of the following statements about Fubini's Theorem is correct?

Accepted Answer

If $$f(x, y) is $$continuous on $$R = [a, b] 	imes [c, d]$$, then $$\iint_R f(x, y)\, dA = \int_a^b \left( \int_c^d f(x, y)\, dy ight) dx = \int_c^d \left( \int_a^b f(x, y)\, dx ight) dy$$

Question 7

A chemical engineer needs to find the average temperature over a rectangular metal plate $$R = [0, 4] 	imes [0, 2]$$ where the temperature at any point is given by $$T(x, y) = 10 + 2x - y. $$What is the average temperature?

Accepted Answer

$$12$$

Question 8

Which of the following is the correct setup for the double integral $$\iint_R x^2$ y\, dA$$ over $$R = [0, 1] 	imes [0, 2]$$?

Accepted Answer

$$\$$int_0$$^1 \$$int_0$$^2 x^2$ y\, dy\, dx$$

Question 9

A rectangular garden $$10$$ meters long and $$5$$ meters wide has a fertilizer concentration at each point $$(x, y)$$ given by $$C(x, y) = 2x + 3y. $$What is the total amount of fertilizer over the garden?

Accepted Answer

$$\displaystyle \iint_R (2x + 3y)\, dA = 875$$

Question 10

Which of the following best explains why the double integral $$\iint_R f(x, y)\, dA$$ gives the volume under $$z = f(x, y)$$ and above the region $$R$$?

Accepted Answer

Because it sums the products of the height $$f(x, y)$$ and the area $$dA$$ over all points in $$R$$, analogous to summing the volumes of infinitesimal columns.

Question 11

If $$f(x, y) = x - 3y$$ and $$R = [0, 2] 	imes [0, 3]$$, what is $$\iint_R f(x, y)\, dA$$?

Accepted Answer

$$-6$$

Question 12

A hospital is monitoring the density of a medication in a rectangular patch of tissue $$R = [0, 2] 	imes [0, 4]$$ with density $$d(x, y) = 5 - x + 2y. $$What is the total amount of medication in the patch?

Accepted Answer

$$52$$

Question 13

Which of the following is NOT a necessary step in setting up a Riemann sum to approximate a double integral over a rectangle?

Accepted Answer

Finding the exact value of the double integral analytically.

Question 14

Evaluate $$\iint_R 1\, dA$$ over $$R = [1, 4] 	imes [2, 5].$$

Accepted Answer

$$9$$

Question 15

If $$f(x, y) is $$continuous on $$R = [a, b] 	imes [c, d]$$, which of the following is always true?

Accepted Answer

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

Question 16

A rectangular field $$R = [0, 5] 	imes [0, 2]$$ has a height profile $$h(x, y) = 3x + 2y. $$What is the total volume of soil above the field if the height is measured from the base?

Accepted Answer

$$95$$

Question 17

Which of the following is the correct order of steps to evaluate $$\iint_R f(x, y)\, dA$$ using Fubini's Theorem?

Accepted Answer

Integrate with respect to $$y$$ first, then with respect to $$x$$, or vice versa, depending on the limits and function.

Double Integrals over Rectangular Regions

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