Question 1

Given the parametric equations $$x = 5 \cos(t)$$ and $$y = 5 \sin(t)$$ for $$0 \leq t \leq \pi$$, what is the surface area generated when this semicircle is rotated about the x-axis?

Accepted Answer

$$S = \$$int_0$$^\pi 2\pi y(t) \sqrt{(x'(t))^2$ + (y'(t))^2$} \, dt$$

Question 2

What is the value of $$x'(t)$$ for the parametric equation $$x = 5 \cos(t)$$?

Accepted Answer

$$x'(t) = -5 \sin(t)$$

Question 3

If $$y = 5 \sin(t)$$, what is $$y'(t)$$?

Accepted Answer

$$y'(t) = 5 \cos(t)$$

Question 4

Which formula correctly represents the surface area $$S$$ generated by rotating a parametric curve $$x = f(t)$$, $$y = g(t)$$ about the x-axis?

Accepted Answer

$$S = \int_a^b 2\pi g(t) \sqrt{(f'(t))^2$ + (g'(t))^2$} \, dt$$

Question 5

For the parametric equations $$x = 5 \cos(t)$$, $$y = 5 \sin(t)$$, $$0 \leq t \leq \pi$$, what is the value of $$\sqrt{(x'(t))^2$ + (y'(t))^2$}$$?

Accepted Answer

$$5$$

Question 6

A cup is formed by rotating the semicircle $$x = 5 \cos(t)$$, $$y = 5 \sin(t)$$, $$0 \leq t \leq \pi$$ about the x-axis. What real-world property does the resulting surface area represent?

Accepted Answer

The area of the outer surface of the cup

Question 7

Suppose you want to find the surface area of a sphere using parametric equations. Which parametric equations would you use?

Accepted Answer

$$x = r \cos(t)$$, $$y = r \sin(t)$$, $$0 \leq t \leq 2\pi$$

Question 8

If a parametric curve is rotated about the y-axis instead of the x-axis, which formula should be used for the surface area?

Accepted Answer

$$S = \int_a^b 2\pi f(t) \sqrt{(f'(t))^2$ + (g'(t))^2$} \, dt$$

Question 9

A medical imaging device uses the surface area formula for parametric equations to model the surface of an organ. If the organ's cross-section is given by $$x = 3 \cos(t)$$, $$y = 2 \sin(t)$$, $$0 \leq t \leq \pi$$, what is the expression for the surface area when rotated about the x-axis?

Accepted Answer

$$S = \$$int_0$$^\pi 2\pi y(t) \sqrt{(x'(t))^2$ + (y'(t))^2$} \, dt$$

Question 10

Which of the following is a necessary condition for using the parametric surface area formula $$S = \int_a^b 2\pi y(t) \sqrt{(x'(t))^2$ + (y'(t))^2$} \, dt$$?

Accepted Answer

The curve must not cross the axis of rotation within $$[a, b]$$

Question 11

A chemical reaction vessel is shaped by rotating the curve $$x = 4 \cos(t)$$, $$y = 2 \sin(t)$$, $$0 \leq t \leq \pi$$ about the x-axis. What is the surface area of the vessel?

Accepted Answer

$$S = \$$int_0$$^\pi 2\pi y(t) \sqrt{(x'(t))^2$ + (y'(t))^2$} \, dt$$

Question 12

If $$x = r \cos(t)$$ and $$y = r \sin(t)$$, $$0 \leq t \leq \pi$$, what is the geometric shape formed when the curve is rotated about the x-axis?

Accepted Answer

A sphere

Question 13

Which step is required before applying the parametric surface area formula to a real-world object?

Accepted Answer

Express the object's boundary as parametric equations

Question 14

A historical artifact is modeled by rotating the curve $$x = 2 \cos(t)$$, $$y = 3 \sin(t)$$, $$0 \leq t \leq \pi$$ about the x-axis. What is the value of $$\sqrt{(x'(t))^2$ + (y'(t))^2$}$$?

Accepted Answer

$$\sqrt{4 \sin^2$(t) + 9 \cos^2$(t)}$$

Question 15

Which of the following best describes the role of $$y(t) in $$the parametric surface area formula for rotation about the x-axis?

Accepted Answer

It represents the radius from the axis of rotation to the curve at each point

Question 16

A designer wants to create a lamp shade by rotating the curve $$x = 2 \cos(t)$$, $$y = 4 \sin(t)$$, $$0 \leq t \leq \pi$$ about the x-axis. What is the correct integral for the surface area?

Accepted Answer

$$S = \$$int_0$$^\pi 2\pi y(t) \sqrt{(x'(t))^2$ + (y'(t))^2$} \, dt$$

Question 17

Why is $$\sqrt{(x'(t))^2$ + (y'(t))^2$}$$ included in the parametric surface area formula?

Accepted Answer

It represents the differential arc length of the curve

Surface Area for Parametric Equations

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