Question 1

Which of the following best describes the area under the curve in the first image, where a car travels at a constant velocity of 10 m/s for 10 seconds?

Accepted Answer

It represents the total distance traveled by the car, which is the product of velocity and time.

Question 2

Given the function $$f(x) = 0.25x^2$ + 1 on $$the interval $$[1, 5]$$, as shown in Figures 2, 3, and 4, which method is used in Figure 2 to approximate the area under the curve?

Accepted Answer

Left endpoint rectangles (left Riemann sum)

Question 3

In Figure 3, which Riemann sum is being used to approximate the area under $$f(x) = 0.25x^2$ + 1$$ from $$x=1 to$$ $$x=5$$?

Accepted Answer

Right endpoint rectangles (right Riemann sum)

Question 4

As the number of rectangles increases in a Riemann sum approximation (as shown in Figure 4), what happens to the approximation of the area under the curve?

Accepted Answer

The approximation becomes more accurate and approaches the exact value of the definite integral.

Question 5

Given the shaded region between $$y = f(x)$$ and $$y = g(x)$$ from $$x = a to$$ $$x = b in $$the fifth image, which integral gives the area of the region $$A$$?

Accepted Answer

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

Question 6

In the sixth image, what is the area $$A$$ bounded by $$f(x) = \frac{1}{2}x + 3$$ and $$g(x) = -x^2$ + 1$$ from $$x = -2 to$$ $$x = 1$$?

Accepted Answer

$$\int_{-2}^1 \left( \frac{1}{2}x + 3 - (-x^2$ + 1) \right) dx$$

Question 7

Refer to the Lorenz curve and the table in the seventh image. What does the point (0.40, 0.11) on the Lorenz curve represent?

Accepted Answer

The bottom 40% of families receive 11% of the total income.

Question 8

In the context of the Lorenz curve, what does a Gini Index of 0 indicate?

Accepted Answer

Absolute equality in income distribution.

Question 9

In the eighth image, which integral would you use to find the total area between $$y = f(x)$$ and $$y = g(x)$$ from $$x = -2 to$$ $$x = 3$$?

Accepted Answer

$$\int_{-2}^3 |f(x) - g(x)| dx$$

Question 10

Suppose the area $$A_1 in $$the eighth image is below the x-axis and $$A_2 is $$above the x-axis. What is the signed area between $$y = f(x)$$ and $$y = g(x)$$ from $$x = -2 to$$ $$x = 3$$?

Accepted Answer

The signed area is $$A_2$$ - $$A_1.$$

Question 11

If a function $$f(x) is $$continuous on $$[a, b]$$, which of the following statements about the definite integral $$\int_a^b f(x) dx is $$true?

Accepted Answer

It represents the net area between the curve $$y = f(x)$$ and the x-axis from $$x = a to$$ $$x = b.$$

Question 12

Which of the following best describes the use of Riemann sums in estimating definite integrals?

Accepted Answer

Riemann sums approximate the area under a curve by summing the areas of rectangles under the curve.

Business Calculus Study Guide: Integration and Applications

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