$$\lim_{n \to \infty} \sum_{i=1}^n \left(1+\frac{1}{x_i}\right) \Delta x$$, where $$x_i = 1 + \frac{4i}{n}$$ and $$\Delta x = \frac{4}{n}$$

$$\int x^n dx = \frac{x$$^{n+1}$$}{n+1} + C$$, for $$n \neq -1$$

Question 1

Which of the following best describes the meaning of a definite integral $$\int_a^b f(x)\,dx$$?

Accepted Answer

It gives the total area under the curve $$f(x)$$ from $$x=a to$$ $$x=b.$$

Question 2

Which of the following expressions represents the limit of a right-endpoint Riemann sum for $$\$$int_1$$^5 \left(1+\frac{1}{x}\right)dx$$?

Accepted Answer

$$\lim_{n 	o \infty} \sum_{i=1}^n \left(1+\frac{1}{x_i}ight) \Delta x$$, where $$x_i = 1 + \frac{4i}{n}$$ and $$\Delta x = \frac{4}{n}$$

Question 3

Suppose $$f(x) is an $$odd function and continuous on $$[-a, a]. $$What is $$\int_{-a}^a f(x)dx$$?

Accepted Answer

$$0$$

Question 4

A biologist measures the mortality rate $$m(t) of $$bacteria over time $$t (in $$hours). What does $$\$$int_4$$^8 m(t)dt$$ represent?

Accepted Answer

The total number of bacteria killed between hours 4 and 8.

Question 5

Which property of definite integrals allows you to write $$\int_a^c f(x)dx = \int_a^b f(x)dx + \int_b^c f(x)dx$$?

Accepted Answer

Additivity over intervals

Question 6

If $$F(x) is an $$antiderivative of $$f(x)$$, what does the Fundamental Theorem of Calculus, Part 2, state?

Accepted Answer

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

Question 7

Given $$y = \$$int_3$$^x t e$$^{4t}$$ dt$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$x e$$^{4x}$$

Question 8

Suppose $$f(x) is $$even and continuous on $$[-a, a]. $$Which of the following is true?

Accepted Answer

$$\int_{-a}^a f(x)dx = 2\$$int_0$$^a f(x)dx$$

Question 9

Which of the following is the correct application of Leibniz's Rule for $$\frac{d}{dx} \int_{g(x)}^{h(x)} f(t)dt$$?

Accepted Answer

$$f(h(x))h'(x) - f(g(x))g'(x)$$

Question 10

Evaluate $$\frac{d}{dx} \int_{x^3$}^{2-x^2$} (t^2$-1)dt.$$

Accepted Answer

$$(2-(x^2))^2$-1)(-2x) - ((x^3$)^2-1)(3x^2$)$

Question 11

Which of the following statements about indefinite integrals is correct?

Accepted Answer

$$\int f(x)dx$$ gives a family of antiderivatives of $$f(x).$$

Question 12

A fish grows according to $$g'(a) = 29e$$^{-a}$$ cm/year. If the fish starts at 1 cm, what is its length after 2 years?

Accepted Answer

$$1 + \$$int_0$$^2 29e$$^{-a}$$da$$

Question 13

Which of the following is a correct basic integration formula?

Accepted Answer

$$\int x^n dx = \frac{x$$^{n+1}$$}{n+1} + C$$, for $$n 
eq -1$$

Question 14

If $$\frac{dy}{dx} = f'(x)$$, what does $$\int f'(x)dx$$ represent?

Accepted Answer

An antiderivative of $$f'(x)$$, which is $$f(x) + C.$$

Question 15

Which of the following is true about the definite integral $$\int_a^a f(x)dx$$?

Accepted Answer

It equals $$0.$$

Question 16

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 17

Which of the following is a correct interpretation of the Fundamental Theorem of Calculus, Part 1?

Accepted Answer

The derivative of the area function $$g(x) = \int_a^x f(t)dt is$$ $$f(x).$$

Question 18

Suppose $$\$$int_0$$^2 f(x)dx = 5$$ and $$\$$int_2$$^4 f(x)dx = 3. $$What is $$\$$int_0$$^4 f(x)dx$$?

Accepted Answer

$$8$$

Definite and Indefinite Integrals: Concepts, Properties, and Applications

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