Question 1

Given the region bounded above by $$y = f(x)$$, below by $$y = g(x)$$, and on the sides by $$x = a$$ and $$x = b$$, which of the following integrals correctly represents the area of the region $$S$$ shown in the second image?

Accepted Answer

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

Question 2

In the first image, what does the height of a typical rectangle represent when approximating the area between two curves?

Accepted Answer

$$f(x_i^*) - g(x_i^*)$$

Question 3

Suppose $$f(x) = x^2$ + 1$$ and $$g(x) = x on$$ $$[0,2]. $$What is the area between these curves?

Accepted Answer

$$\$$int_0$$^2 [(x^2$ + 1) - x] \,dx$$

Question 4

If the region between $$y = e^x$$ and $$y = x$$ from $$x = 0 to$$ $$x = 1 is $$considered, what is the correct setup for the area?

Accepted Answer

$$\$$int_0$$^1 (e^x - x)\,dx$$

Question 5

Which of the following best describes the Riemann sum used to approximate the area between two curves $$y = f(x)$$ and $$y = g(x) on$$ $$[a, b]$$?

Accepted Answer

$$\sum_{i=1}^n [f(x_i^*) - g(x_i^*)]\Delta x$$

Question 6

A region is bounded by $$y = x^2$ and y = 2x$$ - $$x^2. $$What are the $$x-$$values of intersection points?

Accepted Answer

$$x = 0$$ and $$x = 1$$

Question 7

For the region between $$y = \sin x$$ and $$y = \cos x$$ from $$x = 0 to$$ $$x = \frac{\pi}{2}$$, which of the following integrals correctly computes the area?

Accepted Answer

$$\$$int_0$$^{\frac{\pi}{2}} |\sin x - \cos x|\,dx$$

Question 8

When using horizontal slices to find the area between two curves, which variable do you integrate with respect to?

Accepted Answer

$$y$$

Question 9

Given the curves $$x = y + 1$$ and $$x = \frac{1}{2}y^2$ - 3$$, what is the correct integral to find the area between $$y = -2$$ and $$y = 4$$?

Accepted Answer

$$\int_{-2}^4 [(y + 1) - (\frac{1}{2}y^2$ - 3)]\,dy$$

Question 10

A region is bounded by $$y = x^2$ and y = 2x$$ - $$x^2$$ between $$x = 0$$ and $$x = 1. $$What is the area of this region?

Accepted Answer

$$\$$int_0$$^1 ((2x - x^2$) - x^2$)\,dx$$

Question 11

Which of the following statements is TRUE about the area between two curves $$y = f(x)$$ and $$y = g(x) on$$ $$[a, b]$$?

Accepted Answer

The area is always nonnegative.

Question 12

If you are given two curves $$y = f(x)$$ and $$y = g(x)$$, and you want to find the area between them using vertical slices, what must be true about $$f(x)$$ and $$g(x) on$$ $$[a, b]$$?

Accepted Answer

$$f(x) \geq g(x)$$ for all $$x in$$ $$[a, b]$$

Question 13

A real-world application: The cost of producing a product is modeled by $$C(x)$$ and the revenue by $$R(x)$$, both as functions of the number of units $$x. $$What does $$\int_a^b [R(x) - C(x)]\,dx$$ represent?

Accepted Answer

The total profit from $$x = a to$$ $$x = b$$

Question 14

Which of the following is the correct algebraic justification for the area between $$y = f(x)$$ and $$y = g(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

$$\int_a^b f(x)\,dx - \int_a^b g(x)\,dx$$

Question 15

If you are given $$x = f(y)$$ and $$x = g(y)$$, and $$f(y) \geq g(y)$$ for $$c \leq y \leq d$$, what is the area between these curves?

Accepted Answer

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

Applications of Definite Integrals: Areas Between Curves

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