Free Printable AP Calculus BC Integral Formulas

1.

Which of the following does NOT represent the area under the curve f(x) on the interval [a,b]?

a)

$\lim_{n\rightarrow\infty}\sum_{k=1}^nf\left(c_k\right)\Delta x$

b)

$\int_a^bf\left(x\right)dx$

c)

$\int_a^b\left|f\left(x\right)\right|dx$

d)

$\lim_{n\rightarrow\infty}\sum_{k=1}^nf\left(c_k\right)\frac{b-a}{n}$

2.

Which of the following represents Left Rectangular Approximation Method?

a)

$\sum_{x=a}^{b-\Delta x}f\left(x\right)\Delta x$

b)

$\sum_{x=a+\Delta x}^bf\left(x\right)\Delta x$

c)

$\sum_{x=a+\frac{\Delta x}{2}}^{b-\frac{\Delta x}{2}}f\left(x\right)\Delta x$ x

d)

$\sum_{x=a}^bf\left(x\right)\Delta x$

3.

Which of the following represents Right Rectangular Approximation Method?

a)

$\sum_{x=a}^{b-\Delta x}f\left(x\right)\Delta x$

b)

$\sum_{x=a+\Delta x}^bf\left(x\right)\Delta x$

c)

$\sum_{x=a+\frac{\Delta x}{2}}^{b-\frac{\Delta x}{2}}f\left(x\right)\Delta x$ x

d)

$\sum_{x=a}^bf\left(x\right)\Delta x$

4.

Which of the following represents Mid-Point Rectangular Approximation Method?

a)

$\sum_{x=a}^{b-\Delta x}f\left(x\right)\Delta x$

b)

$\sum_{x=a+\Delta x}^bf\left(x\right)\Delta x$

c)

$\sum_{x=a+\frac{\Delta x}{2}}^{b-\frac{\Delta x}{2}}f\left(x\right)\Delta x$ x

d)

$\sum_{x=a}^bf\left(x\right)\Delta x$

5.

Which of the following does NOT represent Trapezoidal Approximation Method?

a)

$\frac{LRAM\ +\ RRAM}{2}$

b)

$\frac{\Delta x}{2}\left(y_0+2y_1+...+2y_{n-1}+y_n\right)$

c)

$\Delta x\left(\frac{1}{2}y_0+y_1+...+y_{n-1}+\frac{1}{2}y_n\right)$

d)

$\frac{\Delta x}{3}\left(y_0+4y_1+2y_2++4y_3+...+4y_{n-1}+y_n\right)$

6.

Which of the following represents Simpson's Rule for approximation?

a)

$\frac{LRAM\ +\ RRAM}{2}$

b)

$\frac{\Delta x}{2}\left(y_0+2y_1+...+2y_{n-1}+y_n\right)$

c)

$\Delta x\left(\frac{1}{2}y_0+y_1+...+y_{n-1}+\frac{1}{2}y_n\right)$

d)

$\frac{\Delta x}{3}\left(y_0+4y_1+2y_2++4y_3+...+4y_{n-1}+y_n\right)$

7.

Which of the following demonstrates the Order of Integration Rule?

a)

$\int_a^bf\left(x\right)dx=-\int_b^af\left(x\right)dx$

b)

$\int_a^af\left(x\right)dx=0$

c)

$\int_a^bkf\left(x\right)dx=k\int_a^bf\left(x\right)dx$

d)

$\int_a^bf\left(x\right)dx+\int_b^cf\left(x\right)dx=\int_a^cf\left(x\right)dx$

8.

Which of the following demonstrates the Zero Rule of Integration?

a)

$\int_a^bf\left(x\right)dx=-\int_b^af\left(x\right)dx$

b)

$\int_a^af\left(x\right)dx=0$

c)

$\int_a^bkf\left(x\right)dx=k\int_a^bf\left(x\right)dx$

d)

$\int_a^bf\left(x\right)dx+\int_b^cf\left(x\right)dx=\int_a^cf\left(x\right)dx$

9.

Which of the following demonstrates the Constant Multiple Rule of Integration?

a)

$\int_a^bf\left(x\right)dx=-\int_b^af\left(x\right)dx$

b)

$\int_a^af\left(x\right)dx=0$

c)

$\int_a^bkf\left(x\right)dx=k\int_a^bf\left(x\right)dx$

d)

$\int_a^bf\left(x\right)dx+\int_b^cf\left(x\right)dx=\int_a^cf\left(x\right)dx$

10.

Which of the following demonstrates the Additivity Rule of Integration?

a)

$\int_a^bf\left(x\right)dx=-\int_b^af\left(x\right)dx$

b)

$\int_a^af\left(x\right)dx=0$

c)

$\int_a^b\left(f\left(x\right)\pm g\left(x\right)\right)dx=\int_a^bf\left(x\right)dx\pm\int_a^bg\left(x\right)dx$

d)

$\int_a^bf\left(x\right)dx+\int_b^cf\left(x\right)dx=\int_a^cf\left(x\right)dx$

11.

Which of the following demonstrates the Sum & Difference Rule of Integration?

a)

$\int_a^bf\left(x\right)dx=-\int_b^af\left(x\right)dx$

b)

$\int_a^af\left(x\right)dx=0$

c)

$\int_a^b\left(f\left(x\right)\pm g\left(x\right)\right)dx=\int_a^bf\left(x\right)dx\pm\int_a^bg\left(x\right)dx$

d)

$\int_a^bf\left(x\right)dx+\int_b^cf\left(x\right)dx=\int_a^cf\left(x\right)dx$

12.

$\min\ f\cdot\left(b-a\right)\le\int_a^bf\left(x\right)dx\le\max\ f\cdot\left(b-a\right)$

a)

Max-Min Inequality Rule of Integration

b)

Domination Rule of Integration

c)

Power Rule of Integration

d)

Mean Value of a Function f(x)

13.

$if\ f\left(x\right)\ \ge\ g\left(x\right)\ on\ \left[a,b\right],\ then\ \int_a^bf\left(x\right)dx\ \ge\int_a^bg\left(x\right)dx$ $if\ f\left(x\right)\ \ge\ 0\ on\ \left[a,b\right],\ then\ \int_a^bf\left(x\right)dx\ \ge0$

a)

Max-Min Inequality Rule of Integration

b)

Domination Rule of Integration

c)

Power Rule of Integration

d)

Mean Value of a Function f(x)

14.

$\int_{ }^{ }u^ndu,\ n\ne-1$

a)

$=\frac{1}{n+1}u^{n+1}+c,\ n\ne1$

b)

$=\ln\left|u\right|+c$

c)

$=n\cdot u^{n-1}+c$

d)

$=\frac{1}{n}u^n+c$

15.

$\ln\left|u\right|+c=$

a)

$\int_{ }^{ }e^udu$

b)

$\int_{ }^{ }\frac{1}{u}du$

c)

$\int_{ }^{ }\ln\left(u\right)\ du$

d)

$\int_{ }^{ }u\ln\left(u\right)du$

16.

Mean Value of a Function f(x) on the interval [a,b]

a)

$\frac{f\left(x\right)}{b-a}$

b)

$\ \frac{\max f}{b-a}$

c)

$\int_a^bf\left(x\right)dx$

d)

$\frac{1}{b-a}\int_a^bf\left(x\right)dx$

17.

Select all of the equations below that are an example of the Fundamental Theorem of Calculus:

a)

$\frac{\text{d}}{\text{d}x}\int_a^xf\left(t\right)dt\ =\ f\left(x\right)$

b)

$\frac{\text{d}}{\text{d}x}\int_a^uf\left(t\right)dt=f\left(u\right)\frac{\text{d}u}{\text{d}x}$

c)

$\int_a^bf\left(x\right)dx=F\left(b\right)-F\left(a\right)$

d)

$\int_{ }^{ }f\left(x\right)dx=F\left(x\right)+c$ where $F'\left(x\right)=f\left(x\right)$

18.

The total area between the curve of f(x) and the x-axis is:

a)

$\int_a^bf\left(x\right)dx$

b)

$\int_a^b\left|f\left(x\right)\right|dx$

c)

$\frac{1}{b-a}\int_a^bf\left(x\right)dx$

d)

$\int_a^b\left|v\left(t\right)\right|dt$

19.

The total distance traveled for the velocity function v(t) from time a to b is:

a)

$\int_a^bv\left(t\right)dt$

b)

$\int_a^b\left|f\left(x\right)\right|dx$

c)

$\frac{1}{b-a}\int_a^bv\left(t\right)dt$

d)

$\int_a^b\left|v\left(t\right)\right|dt$

20.

$\int_{ }^{ }\cos\left(u\right)du=$

a)

$\cos\left(u\right)+c$

b)

$\sin\left(u\right)+c$

c)

$-\sin\left(u\right)+c$

d)

$-\cos\left(u\right)+c$

21.

$\int_{ }^{ }\sin\left(u\right)du=$

a)

$\cos\left(u\right)+c$

b)

$\sin\left(u\right)+c$

c)

$-\sin\left(u\right)+c$

d)

$-\cos\left(u\right)+c$

22.

$\int_{ }^{ }\sec^2\left(u\right)du=$

a)

$\sec\left(u\right)+c$

b)

$\tan\left(u\right)+c$

c)

$-\tan\left(u\right)+c$

d)

$\cot\left(u\right)+c$

23.

$\int_{ }^{ }\csc^2\left(u\right)du=$

a)

$-\csc\left(u\right)+c$

b)

$\cot\left(u\right)+c$

c)

$-\tan\left(u\right)+c$

d)

$-\cot\left(u\right)+c$

24.

$\int_{ }^{ }\sec\left(u\right)\tan\left(u\right)du=$

a)

$\sec\left(u\right)+c$

b)

$\tan\left(u\right)+c$

c)

$-\tan\left(u\right)+c$

d)

$-\sec\left(u\right)+c$

25.

$\int_{ }^{ }\csc\left(u\right)\cot\left(u\right)du=$

a)

$\csc\left(u\right)+c$

b)

$\cot\left(u\right)+c$

c)

$-\cot\left(u\right)+c$

d)

$-\csc\left(u\right)+c$

26.

$\int_{ }^{ }\ \frac{\text{1}}{\sqrt{1-u^2}}du=$

a)

$\sin^{-1}\left(u\right)+c$

b)

$\cos^{-1}\left(u\right)+c$

c)

$\tan^{-1}\left(u\right)+c$

d)

$\sec^{-1}\left(u\right)+c$

27.

$\int_{ }^{ }\ \frac{-\text{1}}{\sqrt{1-u^2}}du=$

a)

$\sin^{-1}\left(u\right)+c$

b)

$\cos^{-1}\left(u\right)+c$

c)

$\cot^{-1}\left(u\right)+c$

d)

$\csc^{-1}\left(u\right)+c$

28.

$\int_{ }^{ }\ \frac{\text{1}}{1+u^2}du=$

a)

$\sin^{-1}\left(u\right)+c$

b)

$\cos^{-1}\left(u\right)+c$

c)

$\tan^{-1}\left(u\right)+c$

d)

$\sec^{-1}\left(u\right)+c$

29.

$\int_{ }^{ }\ \frac{-\text{1}}{1+u^2}du=$

a)

$\cot^{-1}\left(u\right)+c$

b)

$\cos^{-1}\left(u\right)+c$

c)

$\tan^{-1}\left(u\right)+c$

d)

$\csc^{-1}\left(u\right)+c$

30.

$\int_{ }^{ }\ \frac{\text{1}}{\left|u\right|\sqrt{u^2-1}}du=$

a)

$\sin^{-1}\left(u\right)+c$

b)

$\cos^{-1}\left(u\right)+c$

c)

$\csc^{-1}\left(u\right)+c$

d)

$\sec^{-1}\left(u\right)+c$

31.

$\int_{ }^{ }\ \frac{-\text{1}}{\left|u\right|\sqrt{u^2-1}}du=$

a)

$\sin^{-1}\left(u\right)+c$

b)

$\cos^{-1}\left(u\right)+c$

c)

$\csc^{-1}\left(u\right)+c$

d)

$\sec^{-1}\left(u\right)+c$

32.

$\int_{ }^{ }e^udu=$

a)

$e^{u\ }\ln\left|u\right|+c$

b)

$\frac{e^u}{\ln\left|u\right|}+c$

c)

$e^u$

d)

$e^u+c$

33.

$\int_{ }^{ }a^udu=$

a)

$a^{u\ }\ln a+c$

b)

$\frac{a^u}{\ln a}+c$

c)

$a^u$

d)

$a^u+c$

34.

$\int_{ }^{ }\ln\left(u\right)du=$

a)

$\ln\left(u\right)+c$

b)

$\ln\left(u\right)-u+c$

c)

$u\ln\left(u\right)-u+c$

d)

$u\ln\left(u\right)+c$

35.

$\int_{ }^{ }\log_a\left(u\right)du=$

a)

$\frac{\ln\left(x\right)}{\ln\left(a\right)}+c$

b)

$u\log_a\left(u\right)-u+c$

c)

$\frac{u\ln\left(u\right)-u}{\ln a}+c$

d)

$u\log_a\left(u\right)+c$

36.

Which of the following are examples of Integration by Substitution?

a)

$\int_{ }^{ }f'\left(g\left(x\right)\right)g'\left(x\right)dx=f\left(g\left(x\right)\right)+c$

b)

$\int_a^bf'\left(x\right)dx=\int_{u\left(a\right)}^{u\left(b\right)}f'\left(u\right)du=f\left(u\left(b\right)\right)-f\left(u\left(a\right)\right)$

c)

$\int_{ }^{ }f\left(x\right)g'\left(x\right)dx=f\left(x\right)g\left(x\right)-\int_{ }^{ }g\left(x\right)f'\left(x\right)dx$

d)

$\int_{ }^{ }f'\left(x\right)g'\left(x\right)dx=f\left(x\right)g\left(x\right)+c$

37.

Which of the following are examples of Integration by Parts?

a)

$\int_{ }^{ }f'\left(g\left(x\right)\right)g'\left(x\right)dx=f\left(g\left(x\right)\right)+c$

b)

$\int_{ }^{ }udv=uv-\int_{ }^{ }vdu$

c)

$\int_{ }^{ }f\left(x\right)g'\left(x\right)dx=f\left(x\right)g\left(x\right)-\int_{ }^{ }g\left(x\right)f'\left(x\right)dx$

d)

$\int_{ }^{ }f'\left(x\right)g'\left(x\right)dx=f\left(x\right)g\left(x\right)+c$

38.

The following equations are an example of: $\frac{\text{d}y}{\text{d}x}=f\left(y\right)g\left(x\right)$

\int_{ }^{ }\ \frac{1}{f\left(y\right)}dy=\int_{ }^{ }g\left(x\right)dx

a)

Fundamental Theorem of Calculus

b)

Separable Differential Equations

c)

Integration by Parts

d)

Integration by Substitution

39.

$\frac{\text{d}y}{\text{d}x}=ky$ is an example of:

a)

Periodically Compounded Interest

b)

Separable Differential Equations

c)

Exponential Change

d)

Integration by Parts

40.

Which of the following are examples of logistic differential equations?

a)

$\frac{dP}{dt}=kP\left(M-P\right)$

b)

$\frac{dP}{dt}=k\left(M-P\right)$

c)

$P\left(t\right)=\frac{M}{1+Ae^{-Mkt}}$

d)

$P\left(t\right)=\frac{M}{Ae^{-Mkt}}$

41.

The area between the curves of f(x) and g(x):

a)

$\int_a^b\left(f\left(x\right)\ -\ g\left(x\right)\right)dx$

b)

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

c)

$\int_a^b\pi\left(f\left(x\right)\ -\ g\left(x\right)\right)dx$

d)

$\int_a^b\pi\left(f\left(x\right)^2\ -\ g\left(x\right)^2\right)dx$

42.

Select all formulas for the volume of a solid:

a)

$\int_a^bArea\left(x\right)dx$

b)

$\int_a^b\pi\left(f\left(x\right)\right)^2dx$

c)

$\int_a^b\pi\left(\left(f\left(x\right)\right)^2-\left(g\left(x\right)\right)^2\right)dx$

d)

$\int_a^b2\pi r\left(x\right)h\left(x\right)dx$

e)

$\int_a^b\pi\left(f\left(x\right)-g\left(x\right)\right)^2dx$

43.

The length of the curve of f(x) or g(y) from the point (a,c) to (b,d); (select all that apply):

a)

$\int_a^b\sqrt{1+\left(\frac{\text{d}y}{\text{d}x}\right)^2}dx$

b)

$\int_c^d\sqrt{1+\left(\frac{\text{d}x}{\text{d}y}\right)^2}dy$

c)

$\int_a^b\sqrt{1+f"\left(x\right)}dx$

d)

$\int_a^b\sqrt{1+\left(f'\left(x\right)\right)^2}dx$

AP Calculus BC Integral Formulas

Total questions: 43

Worksheet time: 23mins

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