Question 1

The first image shows a 3D surface with a bowl-like shape and vertical stripes. Which of the following equations most likely describes the surface shown in the image?

Accepted Answer

$$z = -x^2$ - y^2$

Question 2

The second image shows a 3D object with circular symmetry about the vertical axis. Which calculus concept is most directly used to compute the volume of such a solid?

Accepted Answer

The Method of Solids of Revolution (Disk or Shell Method)

Question 3

Suppose the surface in the first image is given by $$z = -x^2$ - y^2$. What is the area of the region in the xy$-plane bounded by the circle x^2$ + y^2$ = r^2$?

Accepted Answer

$$\pi r^2$

Question 4

If the bowl-shaped surface in the first image is revolved around the $$z-$$axis, which integral would you use to find the volume enclosed by the surface from $$z=0 to$$ $$z=-h$$?

Accepted Answer

$$V = \$$int_0$$^R \pi (z(r))^2$ dr$$

Question 5

Which of the following best describes the symmetry of the object in the second image?

Accepted Answer

It is symmetric about the $$z-$$axis.

Question 6

If the surface in the first image is described by $$z = -x^2$ - y^2$, what is the gradient $$
abla z$$ at the point (1$$,1)$$?

Accepted Answer

$$\langle -2, -2 \rangle$$ evaluated at $$(1,1)$$, so $$\langle -2, -2 \rangle$$

Question 7

Which of the following integrals would you use to find the surface area of the bowl-shaped surface $$z = -x^2$ - y^2$ over the disk x^2$ + y^2$ \leq r^2$?

Accepted Answer

$$\iint_D \sqrt{1 + (z_x)^2$ + (z_y)^2$}\,dx\,dy$$

Question 8

If the bowl in the first image is a paraboloid $$z = -x^2$ - y^2$, what is the value of z$ at the origin?

Accepted Answer

$$z = 0$$

Question 9

Suppose you want to compute the volume of the solid in the second image using the disk method. Which variable would you integrate with respect to?

Accepted Answer

The radial variable $$r ($$distance from the axis of rotation)

Question 10

If the surface in the first image is $$z = -x^2$ - y^2$, what is the equation of the tangent plane at the point (a$$, b)$$?

Accepted Answer

$$z = -a^2$ - b^2$ - 2a(x-a) - 2b(y-b)$$

Volumes of Solids of Revolution: The Paraboloid Example

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