Question 1

Which of the following integrals correctly represents the area of the region bounded by the curves $$f(x) = x^2$ + 2$$ and $$g(x) = -x$$ between $$x = 0$$ and $$x = 1$$?

Accepted Answer

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

Question 2

Suppose $$f(x)$$ and $$g(x)$$ are continuous on $$[a, b]$$ and $$g(x) \leq f(x)$$ for all $$x in$$ $$[a, b]. $$What is the formula for the area between the two curves from $$x = a to$$ $$x = b$$?

Accepted Answer

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

Question 3

Given the region bounded by $$y = 2 - x^2$ and y = x$, what are the x$-coordinates of their intersection points?

Accepted Answer

$$x = -2$$ and $$x = 1$$

Question 4

Which of the following best describes the process of finding the area between two curves using integration?

Accepted Answer

Accumulating the difference in $$y-$$values between the two curves over the interval

Question 5

If the region between $$y = 4 - x^2$ and the x$-axis is to be found, which integral should be used?

Accepted Answer

$$\int_{-2}^2 (4 - x^2$) dx$$

Question 6

A vertical representative rectangle is used in the area between two curves formula. What does the width $$\Delta x of $$this rectangle represent?

Accepted Answer

A change in the $$x-$$direction

Question 7

In a real-world scenario, the area between two curves could represent which of the following?

Accepted Answer

The difference in profit between two competing companies over time

Question 8

If $$f(x)$$ and $$g(x)$$ intersect at $$x = a$$ and $$x = b$$, and $$f(x) \geq g(x) on$$ $$[a, b]$$, what is the area between the curves?

Accepted Answer

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

Question 9

Which of the following is a necessary condition for using the formula $$A = \int_a^b [f(x) - g(x)] dx to $$find the area between two curves?

Accepted Answer

$$f(x)$$ and $$g(x)$$ must be continuous on $$[a, b]$$

Question 10

A region is bounded by $$y = x^2$ and y = x + 2$. What is the first step to find the area between these curves?

Accepted Answer

Find the points of intersection by solving $$x^2 = x + 2$

Question 11

Which of the following best explains why the area between two curves can be interpreted as an accumulation process?

Accepted Answer

Because the area is the sum of infinitely many infinitesimal differences between the curves

Question 12

If you are given $$f(x) = x^2$ + 2$$ and $$g(x) = -x on$$ $$[0, 1]$$, what is the value of $$f(x) - g(x)$$?

Accepted Answer

$$x^2 + 2 + x$

Question 13

A region is bounded by $$y = 2 - x^2$ and y = x$. What is the correct integral to find the area between these curves?

Accepted Answer

$$\int_{-2}^1 [(2 - x^2$) - x] dx$$

Question 14

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

Accepted Answer

The area is given by $$\int_a^b |f(x) - g(x)| dx if $$the curves cross within $$[a, b]$$

Question 15

In engineering, the area between two curves can be used to calculate which of the following?

Accepted Answer

The work done by a variable force over a distance

Applications of Integration: Area Between Two Curves

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