Question 1

Given the first image, which shows the area between $$y = f(x)$$ and $$y = g(x)$$ from $$x = a to$$ $$x = b$$, what is the correct formula for the area of the shaded region?

Accepted Answer

$$A = \int_a^b [f(x) - g(x)]\,dx$$

Question 2

In the second image, the area is bounded by $$x = f(y)$$ and $$x = g(y)$$ from $$y = c to$$ $$y = d. $$Which integral represents the area of the shaded region?

Accepted Answer

$$A = \int_c^d [f(y) - g(y)]\,dy$$

Question 3

Refer to the third image. If $$y = f(x)$$ and $$y = g(x)$$ intersect at $$x = c$$ between $$x = a$$ and $$x = b$$, how do you compute the total area between the curves from $$x = a to$$ $$x = b$$?

Accepted Answer

$$A = \int_a^c [f(x) - g(x)]\,dx + \int_c^b [g(x) - f(x)]\,dx$$

Question 4

The fourth image shows a solid of revolution formed by rotating the region between $$y = f(x)$$ and $$y = g(x)$$ about the line $$y = a. $$What is the formula for the volume of this solid using the washer method?

Accepted Answer

$$V = \int_{x_1$}^{x_2$} \pi \left[(a - g(x))^2$ - (a - f(x))^2$\right] dx$$

Question 5

In the fifth image, the region between $$y = f(x)$$ and $$y = g(x) is $$revolved about the line $$y = a. $$Which formula gives the volume of the resulting solid?

Accepted Answer

$$V = \int_{x_1$}^{x_2$} \pi \left[(g(x) - a)^2$ - (f(x) - a)^2$\right] dx$$

Question 6

The sixth image shows a solid of revolution formed by rotating the region between $$x = f(y)$$ and $$x = g(y)$$ about the line $$x = a. $$What is the correct formula for the volume using the washer method?

Accepted Answer

$$V = \int_{y_1$}^{y_2$} \pi \left[(f(y) - a)^2$ - (g(y) - a)^2$\right] dy$$

Question 7

In the seventh image, the region between $$x = f(y)$$ and $$x = g(y) is $$revolved about the line $$x = a. $$Which formula gives the volume of the resulting solid?

Accepted Answer

$$V = \int_{y_1$}^{y_2$} \pi \left[(a - f(y))^2$ - (a - g(y))^2$\right] dy$$

Question 8

Suppose you are asked to find the area between $$y = x^2$ and y = x + 2$ from x = 0$ to x = 2$. Which of the following integrals correctly represents this area?

Accepted Answer

$$\$$int_0$$^2 [(x + 2) - x^2$] dx$$

Question 9

A region is bounded by $$y = \sin x$$ and $$y = \cos x$$ between $$x = 0$$ and $$x = \frac{\pi}{4}. $$What is the area of the region?

Accepted Answer

$$\$$int_0$$^{\frac{\pi}{4}} [\cos x - \sin x] dx$$

Question 10

If the region between $$y = x^2$ and y = 4$ is revolved about the x$-axis from x = 0$ to x = 2$, what is the formula for the volume of the solid generated?

Accepted Answer

$$V = \$$int_0$$^2 \pi [4^2$ - (x^2$)^2$$] dx$

Comprehensive Calculus Study Notes: Limits, Derivatives, Integrals, and Applications

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