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 $$x$$ units is $$P'(x) = \sqrt{x} - 6. $$What does the definite integral $$\$$int_0$$^{60} (\sqrt{x} - 6) dx$$ represent in a business context?

Accepted Answer

The total profit from selling 60 units.

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

$$\ln(4) - \ln(1) = \ln(4)$$, 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 $$f$$ with antiderivative $$F$$ 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

$$1$$

Question 10

A company’s weekly revenue (in thousands of dollars) is modeled by $$R(x) = 0.05x^4$ - 1.2x^3$ + 9.6x^2$ - 28.6x + 38.3$$, where $$x is $$the number of weeks since the start of the year. What does the moving average $$R_{av}(x) = \frac{1}{3} \int_{x-3}^x R(t) dt$$ represent?

Accepted Answer

The average weekly revenue over the most recent 3-week period ending at week $$x.$$

Question 11

Evaluate $$\int (3x^2$ + 2x - 5) dx.$$

Accepted Answer

$$x^3$$ + $$x^2 - 5x + C$

Question 12

Which substitution would you use to evaluate $$\int x e$$^{x^2}$$ dx$$?

Accepted Answer

Let $$u = x^2$, so du = 2x dx$

Question 13

Find $$\$$int_0$$^1 (2x + 3) dx.$$

Accepted Answer

$$4$$

Question 14

Suppose $$f(x) \geq g(x)$$ for all $$x in$$ $$[a, b]. $$What does $$\int_a^b [f(x) - g(x)] dx$$ represent?

Accepted Answer

The area between the curves $$f(x)$$ and $$g(x)$$ from $$x = a to$$ $$x = b.$$

Question 15

Evaluate $$\$$int_0$$^2 (3x + 2) dx$$ using the geometric interpretation (area under the line).

Accepted Answer

$$10$$

Question 16

Which of the following is a correct substitution for evaluating $$\int \frac{2x}{x^2$ + 1} dx$$?

Accepted Answer

Let $$u = x^2$ + 1$$, so $$du = 2x dx$$

Question 17

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

Accepted Answer

The average temperature over the 24-hour period.

Question 18

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

Accepted Answer

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

Chapter 4: Integration – Concepts, Properties, and Applications

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