Question 1

Which of the following integrals correctly represents the area between the curves $$y = f(x)$$ and $$y = g(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

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

Question 2

Given the region bounded by $$y = 5 - x^2$ and y$$ = $$x^2 - 3$ from x = -2$ to x = 2$, what is the area of the region?

Accepted Answer

$$\frac{64}{3}$$

Question 3

A region is bounded above by $$y = x + 6$$, below by $$y = 0$$, and on the right by $$y = x^3$. What is the total area of the region from x = -6$ to x = 2$?

Accepted Answer

$$28$$

Question 4

When integrating with respect to $$y to $$find the area between $$x = f(y)$$ and $$x = g(y)$$ from $$y = c to$$ $$y = d$$, which formula should be used?

Accepted Answer

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

Question 5

Find the area of the region bounded by $$x^2 - 2 = x$ and y = x$.

Accepted Answer

$$\frac{9}{2}$$

Question 6

For the region bounded by $$y = 8 - x$$ and $$x = \frac{(y-2)^2$}{3}$$, what are the $$y-$$values of intersection?

Accepted Answer

$$y = -4,\ 5$$

Question 7

Which of the following best describes the 'slice-and-sum' strategy for finding the area between curves?

Accepted Answer

Dividing the region into thin rectangles, summing their areas, and taking the limit as their width approaches zero.

Question 8

Given the region bounded by $$y = \sqrt{2x}$$, $$y = 2x - 12$$, and $$y = 0$$, what is the area of the region?

Accepted Answer

$$\frac{52}{3}$$

Question 9

If the area between two curves is computed as $$\int_a^b [f(x) - g(x)]\,dx$$, 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 10

A region is bounded by $$y = x + 6$$ and $$y = x^3$ for x$ in $$[0, 2]$$. What is the value of $$\$$int_0^2$$ [(x + 6) - $$x^3$$] dx$$?

Accepted Answer

$$10$$

Question 11

Which substitution is used to simplify $$\int_{-4}^{3} (y-2)^2$ dy$$ when integrating with respect to $$y$$?

Accepted Answer

Let $$u = y - 2$$

Question 12

In a real-world scenario, the area between two cost curves $$C_1(x$$)$$ and C_2$(x)$$ over $$x$$ units produced represents:

Accepted Answer

The total difference in cost between the two production methods over the interval.

Question 13

If $$f(x)$$ and $$g(x)$$ intersect at $$x = a$$ and $$x = b$$, and $$f(x) \u003e g(x) on$$ $$[a, b]$$, what is the area between the curves?

Accepted Answer

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

Question 14

Which of the following is a correct step in finding the area between $$y = 8 - x$$ and $$x = \frac{(y-2)^2$}{3}$$?

Accepted Answer

Solve $$8 - y = \frac{(y-2)^2$}{3}$$ for $$y to $$find the limits of integration.

Question 15

If the area between two curves is negative, what does this indicate?

Accepted Answer

The lower curve was subtracted from the upper curve in the wrong order.

Calculus: Regions Between Curves (Applications of Integration)

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