Question 1

Which of the following statements best describes the relationship between antiderivatives and indefinite integrals?

Accepted Answer

An antiderivative of a function $$f(x) is a $$function $$F(x)$$ such that $$F'(x) = f(x)$$, and the indefinite integral $$\int f(x)\,dx$$ represents the family of all such antiderivatives plus a constant.

Question 2

Find the specific antiderivative $$F(x) of$$ $$f(x) = 2x$$ that satisfies $$F(1) = 5.$$

Accepted Answer

$$F(x) = x^2$ + 4$$

Question 3

Which of the following best describes the Fundamental Theorem of Calculus, Part 2?

Accepted Answer

If $$F(x) is $$any antiderivative of $$f(x)$$, then $$\int_a^b f(x)\,dx = F(b) - F(a).$$

Question 4

Suppose the graph of $$f(x) is a $$straight line from $$(0,0) to$$ $$(2,4). $$What is $$\$$int_0$$^2 f(x)\,dx$$?

Accepted Answer

$$\displaystyle 4$$

Question 5

If $$F(x) = \$$int_1$$^x \sin(t^2$)\,dt$$, what is $$F'(x)$$ according to the Fundamental Theorem of Calculus, Part 1?

Accepted Answer

$$\sin(x^2$)$$

Question 6

Which of the following is a property of definite integrals?

Accepted Answer

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

Question 7

Evaluate $$\int 2x\cos(x^2$)\,dx$$ using $$u-$$substitution.

Accepted Answer

$$\sin(x^2$) + C$$

Question 8

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

Accepted Answer

$$e^1$$ - $$e^0$$

Question 9

Find the area between the curves $$y = x^2$ and y = x$ on $$[0,1]$$.

Accepted Answer

$$\$$int_0$$^1 (x - x^2$)\,dx$$

Question 10

Which integral represents the area between $$y = x^2$ and y = 4$ using horizontal slices?

Accepted Answer

$$\$$int_0$$^4 (4 - \sqrt{y})\,dy$$

Question 11

Which of the following integrals represents the volume of a solid with base $$x^2$$ + $$y^2$$ \leq 1$$ and cross-sections perpendicular to the x$-axis that are squares?

Accepted Answer

$$\int_{-1}^1 [2\sqrt{1 - x^2$}]^2\,dx$$

Question 12

Set up the integral for the volume of the solid formed by revolving $$y = x^2$ about the x$-axis from x = 0$ to x = 1$ using the disk method.

Accepted Answer

$$\$$int_0$$^1 \pi (x^2$)^2$$\,dx$

Question 13

Which integral gives the arc length of $$y = \ln(x)$$ from $$x = 1 to$$ $$x = e$$?

Accepted Answer

$$\$$int_1$$^e \sqrt{1 + \left(\frac{1}{x}\$$right)^2$$}\,dx$$

Question 14

Set up the surface area integral for the surface generated by revolving $$y = x^2$ from x = 0$ to x = 1$ about the x$-axis.

Accepted Answer

$$\$$int_0$$^1 2\pi x^2$ \sqrt{1 + (2x)^2$}\,dx$$

Calculus Study Guide: Integrals, Area, Volume, and Related Concepts

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