AP Calculus AB Unit 6: Integration and Accumulation of Change Practice Exam

Assessment for Unit 6: Integration and Accumulation of Change

Estimated Time: 60 minutes

Exam Details

Questions: 21 total (18 multiple choice, 3 short answer)
Time: 60 minutes (recommended)
Topics Covered: Unit 6: Integration and Accumulation of Change
Format: Multiple Choice, Short Answer questions matching real exam difficulty
Scoring: Instant results with detailed explanations

A. Multiple-Choice Questions (18 Questions)

Select the one best answer for each question.

1. Oil is leaking from a storage tank at a rate of $L(t)$ liters per hour, where $t$ is measured in hours. Which of the following best interprets the meaning of $\int_0^4 L(t) \, dt = 15$?

(A)The rate at which oil is leaking increased by 15 liters per hour from $t=0$ to $t=4$.(B)The total amount of oil in the tank decreased by 15 liters during the first 4 hours.(C)The average rate of oil leakage was 15 liters per hour from $t=0$ to $t=4$.(D)There are 15 liters of oil remaining in the tank at $t=4$.

2. [Skill: 1.E | Topic: 6.10] Find the value of $\int \frac{1}{x^2 - 8x + 25} \, dx$.

(A)$\ln|x^2 - 8x + 25| + C$(B)$\frac{1}{3} \arctan\left(\frac{x-4}{3}\right) + C$(C)$\arctan\left(\frac{x-4}{3}\right) + C$(D)$\frac{1}{9} \arctan\left(\frac{x-4}{9}\right) + C$

3. [Skill: 1.E | Topic: 6.10] Evaluate $\int_{0}^{1} \frac{x^2}{x+1} \, dx$.

(A)$\ln 2 - \frac{1}{2}$(B)$\ln 2 + \frac{1}{2}$(C)$\frac{1}{2}$(D)$\ln 2$

4. Consider the following two integrals: $$ I_1 = \int \frac{2x}{x^2+9} \, dx \quad \text{and} \quad I_2 = \int \frac{2}{x^2+9} \, dx $$ Which of the following best describes the techniques required to evaluate $I_1$ and $I_2$?

(A)Both $I_1$ and $I_2$ can be evaluated using $u$-substitution with $u = x^2+9$.(B)Both $I_1$ and $I_2$ result in inverse trigonometric functions.(C)$I_1$ is evaluated using $u$-substitution, while $I_2$ results in an inverse tangent function.(D)$I_1$ results in an inverse tangent function, while $I_2$ is evaluated using $u$-substitution.

5. Which of the following is the most appropriate first step to evaluate the integral $\int \frac{3x^2 - 4}{x+2} \, dx$?

(A)Use $u$-substitution with $u = x+2$.(B)Use $u$-substitution with $u = 3x^2 - 4$.(C)Use polynomial long division to rewrite the integrand as $3x - 6 + \frac{8}{x+2}$.(D)Rewrite the integrand as $3x^2(x+2)^{-1} - 4(x+2)^{-1}$ and integrate term by term.

6. Evaluate the indefinite integral $\int \frac{1}{x^2 - 6x + 13} \, dx$.

(A)$\ln$|x^2 - 6x + 13| + C(B)$\frac{1}{2} \arctan\left$($\frac{x-3}{2}\right$) + C(C)$\arcsin\left$($\frac{x-3}{2}\right$) + C(D)$\frac{1}{2} \ln\left$|$\frac{x-3}{2}\right$| + C

7. A particle moves along a straight line with velocity v(t) , measured in meters per minute. Selected values of v(t) at various times t are given in the table below. | t (minutes) | 0 | 4 | 7 | 12 | | :--- | :--- | :--- | :--- | :--- | | v(t) (m/min) | 0 | 40 | 60 | 50 | Using a trapezoidal sum with the three subintervals indicated by the table, what is the approximation of the total distance traveled by the particle from t = 0 to t = 12 minutes?

(A)420 meters(B)505 meters(C)590 meters(D)1010 meters

8. The function f is continuous and strictly positive on the closed interval [2, 10] . If the graph of f is concave up on the open interval (2, 10) , which of the following statements must be true regarding the approximation of $\int_{2}^{10} $f(x) $\, $dx ?

(A)A Left Riemann sum approximation will be an underestimate.(B)A Right Riemann sum approximation will be an underestimate.(C)A trapezoidal sum approximation will be an underestimate.(D)A trapezoidal sum approximation will be an overestimate.

9. Which of the following limits is equal to the definite integral $$\int_{2}^{6} \ln(x) \, dx$$ ?

(A)$$\lim_{n \to \infty} \sum_{k=1}^{n} \ln\left( 2 + \frac{4k}{n} \right) \frac{1}{n}$$(B)$$\lim_{n \to \infty} \sum_{k=1}^{n} \ln\left( 2 + \frac{4k}{n} \right) \frac{4}{n}$$(C)$$\lim_{n \to \infty} \sum_{k=1}^{n} \ln\left( \frac{4k}{n} \right) \frac{4}{n}$$(D)$$\lim_{n \to \infty} \sum_{k=1}^{n} \ln\left( 2 + \frac{k}{n} \right) \frac{4}{n}$$

10. Let $g$ be the function defined by $g(x) = \int_{1}^{2x} \sqrt{t^2 + 9} \, dt$. What is the value of $g'(2)$?

(A)5(B)10(C)$2\sqrt{13}$(D)$\sqrt{13}$

11. Given that $\int_{-2}^{4} f(x) \, dx = 12$ and $\int_{1}^{4} f(x) \, dx = 4$, what is the value of $\int_{-2}^{1} (f(x) + 2) \, dx$?

(A)6(B)10(C)14(D)18

12. Evaluate the definite integral $\int_{1}^{4} \left( \frac{6}{\sqrt{x}} - 2x \right) dx$.

(A)-9(B)-3(C)3(D)9

13. Selected values for a differentiable function $h$ and its derivative $h'$ are shown in the table below. | $x$ | $h(x)$ | $h'(x)$ | |:---:|:---:|:---:| | 0 | 2 | -3 | | 2 | -5 | 4 | Based on the table, what is the value of $\int_{0}^{2} \left( 3h'(x) + 4x \right) dx$?

(A)-13(B)-7(C)7(D)29

14. [Skill: 1.E | Topic: 6.8] Which of the following is equal to $\int \left( 4\sqrt{x} - \frac{3}{x^2} \right) \, dx$ ?

(A)$\frac{8}{3}x^{3/2} + \frac{3}{x} + C$(B)$\frac{8}{3}x^{3/2} - \frac{3}{x} + C$(C)$2x^{-1/2} + \frac{6}{x^3} + C$(D)$6x^{3/2} + \frac{3}{x} + C$

15. [Skill: 1.D | Topic: 6.8] Let $f$ be a function such that $f'(x) = 3x^2 + \sin(x)$. If $f(0) = 5$, what is $f(x)$ ?

(A)$f(x) = x^3 - \cos(x) + 5$(B)$f(x) = x^3 - \cos(x) + 6$(C)$f(x) = x^3 + \cos(x) + 4$(D)$f(x) = x^3 + \cos(x) + 5$

16. Find the indefinite integral $\int x(3x^2 + 2)^4 \, dx$.

(A)$\frac{1}{5}(3x^2 + 2)^5 + C$(B)$\frac{1}{10}(3x^2 + 2)^5 + C$(C)$\frac{1}{30}(3x^2 + 2)^5 + C$(D)$\frac{1}{6}(3x^2 + 2)^5 + C$

17. Which of the following is equivalent to the definite integral $\int_{0}^{\frac{pi}{2}} \sin(x) \cos^3(x) \, dx$ using the substitution $u = \cos(x)$?

(A)$\int_{0}^{1} u^3 \, du$(B)$-\int_{0}^{1} u^3 \, du$(C)$\int_{1}^{0} u^3 \, du$(D)$\int_{0}^{\frac{pi}{2}} u^3 \, du$

18. Which of the following definite integrals is equivalent to the limit given by $\displaystyle \lim_{n \to \infty} \sum_{k=1}^{n} \sqrt{4 + \frac{3k}{n}} \cdot \frac{3}{n}$ ?

(A)$\displaystyle \int_{4}^{7} \sqrt{x} \, dx$(B)$\displaystyle \int_{0}^{3} \sqrt{x} \, dx$(C)$\displaystyle \int_{4}^{7} \sqrt{4+x} \, dx$(D)$\displaystyle \int_{0}^{4} \sqrt{3+x} \, dx$

B. Short Answer Questions (3 Questions)

Answer all parts of each question. Answers must be in essay form. Outlines or lists alone are not acceptable.

Question 19:

Question 20:

Question 21:

加载中...