$$\int_{0}^{2\pi} \int_{0}^{1} r^3$ \, dr \, d\theta$$

$$\int_{0}^{2\pi} \int_{0}^{2} r \, dr \, d\theta$$

$$\int_{0}^{2\pi} \int_{0}^{3} r \, dr \, d\theta$$

Question 1

Which of the following equations best represents the surface shown in Figure 1 (a double cone)?

Accepted Answer

$$x^2$$ + $$y^2$$ = $$z^2$$

Question 2

Figure 2 shows a surface with circular cross-sections and vertical sides. Which equation describes this surface?

Accepted Answer

$$x^2$$ + $$y^2 = 1$

Question 3

Which figure represents a hyperboloid of one sheet?

Accepted Answer

Figure 3

Question 4

Given the equation $$x^2$$ + $$y^2$$ + $$z^2 = 1$, what type of surface does this represent?

Accepted Answer

A sphere

Question 5

Evaluate the double integral $$\int \int_{R} 1 \, dA$$ over the region $$R$$ bounded by $$x^2$$ + $$y^2 = 1$.

Accepted Answer

$$\pi$$

Question 6

Which of the following is the volume of the region bounded by $$x^2$$ + $$y^2 = 1$ and 0$$ \leq z \leq 5$$?

Accepted Answer

$$5\pi$$

Question 7

If $$f(x, y) = x^2$ + y^2$, what is the value of f(2$$, -3)$$?

Accepted Answer

$$13$$

Question 8

Which of the following integrals represents the volume under $$z = x^2$ + y^2$ over the region x^2$ + y^2$ \leq 1$$?

Accepted Answer

$$\int_{0}^{2\pi} \int_{0}^{1} r^3$ \, dr \, d	heta$$

Question 9

What is the value of $$\int_{0}^{1} x^2$ \, dx$$?

Accepted Answer

$$\frac{1}{3}$$

Question 10

Which of the following is the equation of a hyperboloid of two sheets?

Accepted Answer

$$\frac{x^2$}{a^2$} + \frac{y^2$}{b^2$} - \frac{z^2$}{c^2$} = -1$$

Question 11

If the region $$R is $$bounded by $$x = 0$$, $$x = 2$$, $$y = 0$$, and $$y = 3$$, what is the area of $$R$$?

Accepted Answer

$$6$$

Question 12

Which of the following is the correct setup for the double integral to find the area inside $$x^2$$ + $$y^2 = 4$?

Accepted Answer

$$\int_{0}^{2\pi} \int_{0}^{2} r \, dr \, d	heta$$

Question 13

Which of the following is the correct value for $$\int_{0}^{2} x \, dx$$?

Accepted Answer

$$2$$

Question 14

If $$z = x^2$ - y^2$, what is the value of z$ at (x$$, y) = (3, 2)$$?

Accepted Answer

$$5$$

Question 15

Which of the following surfaces is NOT a quadric surface?

Accepted Answer

$$z = e$$^{x+y}$$

Question 16

Which of the following is the correct value for $$\int_{0}^{1} 3x \, dx$$?

Accepted Answer

$$\frac{3}{2}$$

Question 17

Which of the following equations describes a cylinder oriented along the $$z-$$axis?

Accepted Answer

$$x^2$$ + $$y^2$$ = $$r^2$$

Question 18

If $$f(x, y) = x^2$ - y^2$, what is f(-2$$, 1)$$?

Accepted Answer

$$5$$

Question 19

Which of the following is the correct setup for the double integral to find the area inside $$x^2$$ + $$y^2 = 9$?

Accepted Answer

$$\int_{0}^{2\pi} \int_{0}^{3} r \, dr \, d	heta$$

Surfaces and Double Integrals in Multivariable Calculus

Study Guide - Practice Questions