Q: Which of the following is the correct formula for the volume element $$dV in $$cylindrical coordinates?

$$dV = r \, dr \, d\theta \, dz$$

Q: Given the transformation from rectangular to cylindrical coordinates, which of the following is correct?

$$x = r \cos \theta, \quad y = r \sin \theta, \quad z = z$$

Q: A solid cylinder of height $$h$$ and radius $$a is $$centered along the $$z-$$axis. Which of the following sets of limits correctly describes the region in cylindrical coordinates?

$$0 \leq r \leq a, \quad 0 \leq \theta \leq 2\pi, \quad 0 \leq z \leq h$$

Q: Which of the following is the correct formula for the volume element $$dV in $$spherical coordinates?

$$dV = \$$rho^2$$ \sin \phi \, d\rho \, d\phi \, d\theta$$

Q: Convert the point $$(x, y, z) = (0, 1, 0) to $$spherical coordinates $$(\rho, \theta, \phi). $$Which of the following is correct?

$$\rho = 1, \theta = \frac{\pi}{2}, \phi = \frac{\pi}{2}$$

Q: A region is bounded by $$x^2$$ + $$y^2$$ \leq 4$$ and 0$$ \leq z \leq 3$$. What are the limits for r$, $$\theta$$, and z$ in cylindrical coordinates for this region?

$$0 \leq r \leq 2, \quad 0 \leq \theta \leq 2\pi, \quad 0 \leq z \leq 3$$

Q: Which of the following is the correct transformation from spherical to rectangular coordinates?

$$x = \rho \sin \phi \cos \theta, \quad y = \rho \sin \phi \sin \theta, \quad z = \rho \cos \phi$$

Q: Suppose you are integrating a function $$f(x, y, z)$$ over a sphere of radius $$R$$ centered at the origin. Which of the following is the correct set of limits in spherical coordinates?

$$0 \leq \rho \leq R, \quad 0 \leq \theta \leq 2\pi, \quad 0 \leq \phi \leq \pi$$

Question 1

Which of the following is the correct formula for the volume element $$dV in $$cylindrical coordinates?

Accepted Answer

$$dV = r \, dr \, d	heta \, dz$$

Question 2

Given the transformation from rectangular to cylindrical coordinates, which of the following is correct?

Accepted Answer

$$x = r \cos 	heta, \quad y = r \sin 	heta, \quad z = z$$

Question 3

A solid cylinder of height $$h$$ and radius $$a is $$centered along the $$z-$$axis. Which of the following sets of limits correctly describes the region in cylindrical coordinates?

Accepted Answer

$$0 \leq r \leq a, \quad 0 \leq 	heta \leq 2\pi, \quad 0 \leq z \leq h$$

Question 4

Which of the following is the correct formula for the volume element $$dV in $$spherical coordinates?

Accepted Answer

$$dV = \$$rho^2$$ \sin \phi \, dho \, d\phi \, d	heta$$

Question 5

Convert the point $$(x, y, z) = (0, 1, 0) to $$spherical coordinates $$(ho, 	heta, \phi). $$Which of the following is correct?

Accepted Answer

$$ho = 1, 	heta = \frac{\pi}{2}, \phi = \frac{\pi}{2}$$

Question 6

Which of the following correctly expresses $$z in $$terms of spherical coordinates?

Accepted Answer

$$z = \rho \cos \phi$$

Question 7

A region is bounded by $$x^2$$ + $$y^2$$ \leq 4$$ and 0$$ \leq z \leq 3$$. What are the limits for r$, $$	heta$$, and z$ in cylindrical coordinates for this region?

Accepted Answer

$$0 \leq r \leq 2, \quad 0 \leq 	heta \leq 2\pi, \quad 0 \leq z \leq 3$$

Question 8

Which of the following is the correct transformation from spherical to rectangular coordinates?

Accepted Answer

$$x = ho \sin \phi \cos 	heta, \quad y = ho \sin \phi \sin 	heta, \quad z = ho \cos \phi$$

Question 9

Suppose you are integrating a function $$f(x, y, z)$$ over a sphere of radius $$R$$ centered at the origin. Which of the following is the correct set of limits in spherical coordinates?

Accepted Answer

$$0 \leq ho \leq R, \quad 0 \leq 	heta \leq 2\pi, \quad 0 \leq \phi \leq \pi$$

Question 10

Which of the following integrals correctly computes the volume of a right circular cylinder of radius $$a$$ and height $$h$$ using cylindrical coordinates?

Accepted Answer

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

Question 11

A medical imaging device scans a spherical region of the brain with radius $$5 cm. $$What is the correct triple integral in spherical coordinates to compute the total volume scanned?

Accepted Answer

$$\int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^{5} \$$rho^2$$ \sin \phi \, dho \, d\phi \, d	heta$$

Question 12

Which of the following is NOT a correct relationship between rectangular and cylindrical coordinates?

Accepted Answer

$$z = r \sin 	heta$$

Question 13

A chemical reaction occurs within a cylindrical tube of radius $$2 cm $$and height $$10 cm. $$What is the correct setup for the triple integral in cylindrical coordinates to find the total amount of a substance with density $$\delta(r, 	heta, z)$$?

Accepted Answer

$$\int_{0}^{2\pi} \int_{0}^{2} \int_{0}^{10} \delta(r, 	heta, z) \, r \, dz \, dr \, d	heta$$

Question 14

Which of the following is the correct relationship for $$r in $$spherical coordinates?

Accepted Answer

$$r = \rho \sin \phi$$

Question 15

A region in space is described by $$0 \leq ho \leq 3$$, $$0 \leq 	heta \leq 2\pi$$, $$0 \leq \phi \leq \frac{\pi}{2}. $$What is the shape of this region?

Accepted Answer

The upper hemisphere of a sphere of radius $$3$$

Question 16

Which of the following is the correct order of integration for the volume of a sphere in spherical coordinates?

Accepted Answer

$$dho$$, $$d\phi$$, $$d	heta$$

Question 17

A solid bounded by $$z = 0$$ and $$z = \sqrt{4 - x^2$ - y^2$} is $$best described in which coordinate system, and what are the limits for the radial variable?

Accepted Answer

Cylindrical coordinates, $$0 \leq r \leq 2$$

Coordinate Systems in Multivariable Calculus: Cylindrical and Spherical Coordinates

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