Question 1

Which of the following best describes the meaning of the constant of integration $$C in $$the general antiderivative $$F(x) + C of a $$function $$f(x)$$?

Accepted Answer

It is an arbitrary constant that accounts for all possible vertical shifts of the antiderivative.

Question 2

Evaluate the indefinite integral $$\int 7x^2$ dx.$$

Accepted Answer

$$\frac{7}{3}x^3$ + C$$

Question 3

A car travels at a constant velocity of $$60$$ miles per hour for $$1.5$$ hours. Using the concept of area under the velocity-time graph, what is the total distance traveled?

Accepted Answer

$$90$$ miles

Question 4

Which rule allows you to factor a constant out of an indefinite integral?

Accepted Answer

Constant Multiplier Rule

Question 5

Find $$\int e$$^{4x}$$ dx.$$

Accepted Answer

$$\frac{1}{4}e$$^{4x}$$ + C$$

Question 6

Suppose the marginal profit for selling $$x$$ seats on a flight is $$P'(x) = \sqrt{x} - 6 (in $$hundreds of dollars per seat). What does the definite integral $$\$$int_0$$^{60} (\sqrt{x} - 6) dx$$ represent in this context?

Accepted Answer

The total profit (in hundreds of dollars) from selling up to 60 seats.

Question 7

Evaluate the definite integral $$\$$int_1$$^4 \frac{1}{x} dx$$ and interpret its meaning if $$y = \frac{1}{x}$$ represents profit per mile (in dollars) for $$x$$ thousands of miles traveled.

Accepted Answer

$$\ln(4) - \ln(1)$$, representing the total profit from 1,000 to 4,000 miles.

Question 8

Which of the following is the correct application of the Fundamental Theorem of Calculus for a continuous function $$f(x)$$ with antiderivative $$F(x)$$ over $$[a, b]$$?

Accepted Answer

$$\int_a^b f(x) dx = F(b) - F(a)$$

Question 9

Find the average value of $$f(x) = x^2$ on the interval $$[0, 2]$$.

Accepted Answer

$$\frac{4}{3}$$

Question 10

A company’s weekly revenue (in thousands of dollars) is modeled by $$R(x) = 0.05x^4$ - 1.2x^3$ + 9.6x^2$ - 28.6x + 38.3$$ for $$0 \leq x \leq 10. $$What does the moving average $$R_{av}(x)$$ over a 3-week interval represent?

Accepted Answer

The average weekly revenue over each 3-week period.

Question 11

Evaluate $$\$$int_0$$^1 x e$$^{x^2}$$ dx$$ using substitution.

Accepted Answer

$$\frac{1}{2}(e$$^{1}$$ - e$$^{0}$$)$$

Question 12

Which of the following integrals requires the use of substitution to evaluate?

Accepted Answer

$$\int (2x + 1)^5$ dx$$

Question 13

Suppose $$f(x) = 2x + 1$$ and $$g(x) = x^2$ + 1. $$What is the area between $$f(x)$$ and $$g(x)$$ from $$x = 0 to$$ $$x = 2$$?

Accepted Answer

$$\$$int_0$$^2 [f(x) - g(x)] dx$$

Question 14

Evaluate $$\int \frac{2x}{x^2$ + 1} dx.$$

Accepted Answer

$$\ln|x^2$ + 1| + C$$

Question 15

A temperature function is given by $$T(x) = -0.012x^3$ + 0.38x^2$ - 1.99x + 10.1$$ for $$0 \leq x \leq 24. $$What is the formula for the average temperature over this 24-hour period?

Accepted Answer

$$\frac{1}{24} \$$int_0$$^{24} T(x) dx$$

Question 16

Evaluate $$\$$int_0$$^2 (3x + 2) dx$$ using geometry.

Accepted Answer

The area is $$4 ($$rectangle) plus $$6 ($$triangle), so $$10.$$

Question 17

Which of the following expresses the additive property of definite integrals?

Accepted Answer

$$\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$$

Question 18

Evaluate $$\$$int_0$$^1 (2x + 3)^2$ dx$$ using substitution.

Accepted Answer

$$\frac{19}{3}$$

Chapter 4: Integration – Comprehensive Study Notes

Guida di studio - Domande di pratica

Study guide - Flashcard