Q: Which of the following is the correct definition of the definite integral as a limit?

$$\int_a^b f(x)dx = \lim_{n \to \infty} L_n(f) = \lim_{n \to \infty} U_n(f)$$

Question 1

Which of the following best describes the meaning of the definite integral $$\int_a^b f(x)dx$$ when $$f(x) \geq 0$$ for all $$x in$$ $$[a, b]$$?

Accepted Answer

It is the area under the graph of $$f$$ and above the $$x-$$axis from $$x=a to$$ $$x=b.$$

Question 2

Suppose $$f(x) = 1$$ for all $$x in$$ $$[a, b]. $$What is $$\int_a^b f(x)dx$$?

Accepted Answer

$$b-a$$

Question 3

Which property of definite integrals allows you to write $$\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$$ for $$c in$$ $$[a, b]$$?

Accepted Answer

Additivity over intervals

Question 4

If $$\$$int_2$$^5 f(x)dx = 7$$ and $$\$$int_5$$^8 f(x)dx = -3$$, what is $$\$$int_2$$^8 f(x)dx$$?

Accepted Answer

$$4$$

Question 5

Given $$\int_a^b f(x)dx = 5$$ and $$\int_a^b g(x)dx = -2$$, what is $$\int_a^b [f(x) + g(x)]dx$$?

Accepted Answer

$$3$$

Question 6

If $$f(x) is $$continuous on $$[a, b]$$, which of the following is always true?

Accepted Answer

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

Question 7

Suppose $$f(x) \leq g(x)$$ for all $$x in$$ $$[a, b]. $$Which of the following must be true?

Accepted Answer

$$\int_a^b f(x)dx \leq \int_a^b g(x)dx$$

Question 8

A company’s marginal cost function is $$C'(x) = 5. $$What does $$\int_{100}^{200} C'(x)dx$$ represent in this context?

Accepted Answer

The change in total cost when production increases from 100 to 200 units.

Question 9

If $$f(x) \leq M$$ for all $$x in$$ $$[a, b]$$, which of the following inequalities is correct?

Accepted Answer

$$\int_a^b f(x)dx \leq M(b-a)$$

Question 10

Evaluate $$\int_{-2}^3 f(x)dx$$ given that the area above the $$x-$$axis from $$x=-2 to$$ $$x=2 is$$ $$3$$ and the area below the $$x-$$axis from $$x=2 to$$ $$x=3 is$$ $$\frac{1}{2}.$$

Accepted Answer

$$\frac{5}{2}$$

Question 11

Which of the following statements is true about the lower and upper sums $$L_n(f)$$ and $$U_n(f)$$ for a continuous function $$f on$$ $$[a, b]$$?

Accepted Answer

$$L_n(f) \leq U_n(f)$$ for all n$

Question 12

If $$f(x) \leq m$$ for all $$x in$$ $$[a, b]$$, what is the best upper estimate for $$\int_a^b f(x)dx$$?

Accepted Answer

$$m(b-a)$$

Question 13

Which of the following is NOT a property of definite integrals?

Accepted Answer

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

Question 14

If $$f(x) \leq g(x)$$ for all $$x in$$ $$[a, b]$$, which of the following is always true?

Accepted Answer

$$\int_a^b f(x)dx \leq \int_a^b g(x)dx$$

Question 15

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

Accepted Answer

The negative of the area between the graph of $$f$$ and the $$x-$$axis.

Question 16

If $$\$$int_0$$^4 f(x)dx = 10$$ and $$\$$int_0$$^4 g(x)dx = -6$$, what is $$\$$int_0$$^4 [2f(x) - 3g(x)]dx$$?

Accepted Answer

$$32$$

Question 17

Which of the following is the correct definition of the definite integral as a limit?

Accepted Answer

$$\int_a^b f(x)dx = \lim_{n 	o \infty} L_n(f) = \lim_{n 	o \infty} U_n(f)$$

Definite Integrals: Properties, Area Interpretation, and Examples

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