Question 1

Suppose you want to find the area of the region bounded by $$y = 5 - x^2$ and y$$ = $$x^2 - 3$ between their points of intersection. Which of the following integrals correctly represents this area?

Accepted Answer

$$\int_{-2}^{2} \left[(5 - x^2$) - (x^2$ - 3)\right] dx$$

Question 2

Given two continuous functions $$f(x)$$ and $$g(x) on$$ $$[a, b]$$ with $$f(x) \geq g(x)$$, what is the general formula for the area between the curves?

Accepted Answer

$$A = \int_{a}^{b} [f(x) - g(x)] dx$$

Question 3

In which scenario is it more convenient to integrate with respect to $$y$$ rather than $$x$$ when finding the area between two curves?

Accepted Answer

When the region is bounded by curves of the form $$x = f(y)$$ and $$x = g(y)$$

Question 4

A region is bounded by $$y = x^3$, y = x + 6$, and the x$-axis. Which of the following expressions gives the area of this region?

Accepted Answer

$$\int_{a}^{b} [(x + 6) - x^3$] dx$$

Question 5

If $$f(x) \geq g(x) \geq 0 on$$ $$[a, b]$$, how can the area between the curves be interpreted using two separate integrals?

Accepted Answer

$$A = \int_{a}^{b} f(x) dx - \int_{a}^{b} g(x) dx$$

Question 6

Consider the region bounded by $$y = \cos x$$ and $$y = \sin x on$$ $$\left[0, \frac{\pi}{2}\right]. $$What is the area of this region?

Accepted Answer

$$\int_{0}^{\frac{\pi}{2}} [\cos x - \sin x] dx$$

Question 7

Two runners start at the same location and run for 1 hour. Their velocities are $$v_1(t) = 7t$ and v_2$(t) = 10t. $$What does the area between their velocity curves from $$t = 0 to$$ $$t = 1$$ represent?

Accepted Answer

The difference in the total distance each runner travels in 1 hour

Question 8

Which of the following is the correct setup for the area between $$x = y^2$ - 4$$ and $$y = \frac{x}{3}$$?

Accepted Answer

$$\int_{c}^{d} [(y^2$ - 4) - 3y] dy$$

Question 9

Suppose $$f(x)$$ and $$g(x)$$ are even functions on $$[-a, a]$$ with $$f(x) \u003e g(x) \u003e 0. If $$the area between their graphs on $$[-a, a] is 10$$, what is $$\int_{0}^{a} x [f(x^2$) - g(x^2$)] dx$$?

Accepted Answer

5

Question 10

A Lorenz curve $$y = L(x) is $$always below the line $$y = x$$ except at $$x = 0$$ and $$x = 1. $$What does the area between $$y = x$$ and $$y = L(x)$$ represent in economics?

Accepted Answer

A measure of income inequality (the Gini index)

Question 11

Given the region in the first quadrant bounded by $$y = x$$^{2/3}$$ and y = x - 4$, what is the correct integral to find the area?

Accepted Answer

$$\int_{a}^{b} [(x - 4) - x$$^{2/3}$$] dx$$

Question 12

If the area between $$y = f(x)$$ and $$y = g(x) on$$ $$[a, b] is$$ $$A$$, what is the area between $$x = f(y)$$ and $$x = g(y) on$$ $$[c, d] if$$ $$f$$ and $$g$$ are the same functions but with variables swapped?

Accepted Answer

The area is the same, $$A$$

Question 13

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

Accepted Answer

Partition the interval, approximate with rectangles, sum their areas, and take the limit as the number of rectangles increases

Question 14

For the region bounded by $$y = 2 - |x|$$ and $$y = x^2$, which of the following integrals gives the area in the first quadrant?

Accepted Answer

$$\int_{0}^{1} [(2 - x) - x^2$] dx$$

Question 15

If $$A = \int_{a}^{b} [f(x) - g(x)] dx$$, what happens to $$A if$$ $$f(x)$$ and $$g(x)$$ are swapped?

Accepted Answer

$$A$$ changes sign

Question 16

Which of the following is a correct interpretation of the definite integral $$\int_{a}^{b} [f(x) - g(x)] dx$$ when $$f(x) \geq g(x) on$$ $$[a, b]$$?

Accepted Answer

It represents the area of the region between $$y = f(x)$$ and $$y = g(x)$$ from $$x = a to$$ $$x = b$$

Regions Between Curves: Area Calculation Using Integration

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