Question 1

Which of the following best represents the general Riemann sum for approximating the area under the curve $$f(x) on$$ $$[a, b]$$?

Accepted Answer

$$\sum_{i=1}^{n} f(x_i^*) \Delta x$$

Question 2

Given $$\int_{a}^{b} f(x) dx = F(b) - F(a)$$, which theorem is being applied?

Accepted Answer

Fundamental Theorem of Calculus

Question 3

Evaluate $$\int \sin(2x) dx.$$

Accepted Answer

$$-\frac{1}{2} \cos(2x) + C$$

Question 4

Which of the following is always true for a continuous function $$f on$$ $$[a, b] if$$ $$f(x) \geq 0$$ for all $$x in$$ $$[a, b]$$?

Accepted Answer

$$\int_{a}^{b} f(x) dx \geq 0$$

Question 5

Suppose $$f(x) is $$continuous and increasing on $$[a, b]. $$Which Riemann sum method will always overestimate the definite integral?

Accepted Answer

Right-hand sum

Question 6

Find $$\int \frac{1}{1 + (2x)^2$} dx.$$

Accepted Answer

$$\frac{1}{2} \arctan(2x) + C$$

Question 7

If $$F(x) = \int_{1}^{x^2$} e$$^{t^2}$$ dt$$, what is $$F'(x)$$?

Accepted Answer

$$2x e$$^{x^4}$$

Question 8

Which of the following is the correct formula for the average value of a continuous function $$f on$$ $$[a, b]$$?

Accepted Answer

$$\frac{1}{b-a} \int_{a}^{b} f(x) dx$$

Question 9

Evaluate $$\int \sec^2$(x) dx.$$

Accepted Answer

$$	an(x) + C$$

Question 10

A particle moves with velocity $$v(t) = 5t^2$ + 2t - 6. $$What is the total distance traveled from $$t = 1 to$$ $$t = 3$$?

Accepted Answer

$$\int_{1}^{3} (5t^2$ + 2t - 6) dt$$

Question 11

Which property of definite integrals is illustrated by $$\int_{a}^{b} f(x) dx = -\int_{b}^{a} f(x) dx$$?

Accepted Answer

Reversal of limits

Question 12

Evaluate $$\int \frac{1}{\sqrt{1 - (2x)^2$}} dx.$$

Accepted Answer

$$\frac{1}{2} \arcsin(2x) + C$$

Question 13

Which of the following is the correct expression for the midpoint Riemann sum for $$f(x) on$$ $$[a, b]$$ with $$n$$ subintervals?

Accepted Answer

$$\sum_{i=1}^{n} f(x_i^*) \Delta x$$

Question 14

If $$f(x) is $$continuous on $$[a, b]$$ and $$f(x) \leq g(x)$$ 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

Evaluate $$\int x^n dx$$ for $$n 
eq -1.$$

Accepted Answer

$$\frac{x$$^{n+1}$$}{n+1} + C$$

Question 16

Which of the following is the correct formula for the definite integral of $$f(x)$$ from $$a to$$ $$b$$ using the Fundamental Theorem of Calculus?

Accepted Answer

$$F(b) - F(a)$$, where $$F is $$any antiderivative of $$f$$

Question 17

Evaluate $$\int \sec(x) dx.$$

Accepted Answer

$$\ln|\sec(x) + 	an(x)| + C$$

Question 18

If $$f(x) is $$continuous on $$[a, b]$$ and $$f(x) \u003c 0$$ for all $$x in$$ $$[a, b]$$, which of the following is always true?

Accepted Answer

$$\int_{a}^{b} f(x) dx \u003c 0$$

Question 19

Which of the following is the correct formula for the sum $$\sum_{k=1}^{n} k^2$?

Accepted Answer

$$\frac{n(n+1)(2n+1)}{6}$$

Calculus II: Definite Integrals, Riemann Sums, and Applications – Midterm #1 Review

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