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AP Calculus AB : AP Calculus AB

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Example Questions

Example Question #1111 : Ap Calculus Ab

Solve using the trapezoidal approximation:

$\int_{0}^{2}(x^5+e^x)dx$

Possible Answers:

$\frac{33+e^2}{2}$

33+e^2

32+e^2

16+e^2

Correct answer:

33+e^2

Explanation:

The trapezoidal approximation of a definite integral is given by the following:

$\int_{a}^{b}f(x)dx \approx \frac{f(b)+f(a)}{2}(b-a)$

Using this approximation for our integral, we get

$(2-0)(\frac{32+e^2+1}{2})=33+e^2$

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Example Question #1112 : Ap Calculus Ab

Solve the integral using the trapezoidal approximation:

$\int_{0}^{4}xe^{2x}dx$

Possible Answers:

8e^8

Correct answer:

8e^8

Explanation:

To approximate the definite integral using the trapezoidal rule, we use the following approximation:

$\int_{a}^{b}f(x)dx\approx (b-a)(\frac{f(b)+f(a)}{2})$

For our integral, we get

$\int_{0}^{4}xe^{2x}dx \approx (4-0)(\frac{4e^8}{2}+0)=8e^8$

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Example Question #1113 : Ap Calculus Ab

Approximate the integral using the trapezoidal rule:

$\int_{0}^{6}x^2\sin(\frac{\pi x}{12})dx$

Possible Answers:

216

108

Correct answer:

108

Explanation:

To approximate the definite integral using the trapezoidal rule, we use the following approximation:

$\int_{a}^{b}f(x)dx\approx (b-a)(\frac{f(b)+f(a)}{2})$

For our integral, we get

$\int_{0}^{6}x^2\sin(\frac{\pi x}{12})dx \approx \frac{(6)(36+0)}{2}=108$

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Example Question #41 : Trapezoidal Sums

Using the Trapezoidal Rule with four subintervals, approximate to four decimal places:

$\int_{2}^{3} \log_{3}( x^{2} ) \mathrm{d} x$

Possible Answers:

1.6558

1.5620

1.7465

2.0620

1.6542

Correct answer:

1.6542

Explanation:

The Trapezoidal Rule states that a definite integral

$\int_{a}^{b} f(x) \textup{d}x$

can be approximated by computing the expression

$T_{N}=\frac{b-a}{2N} \left [f (x_{0} ) +2 f (x_{1}) +2 f ( x_{2} ) + ... 2f ( x_{N-1} )+ f (x_{N} ) \right ]$

where $x_{k} = a + \frac{b-a}{N} \cdot k$ for k = 0, 1, 2, 3, ...,N

Since we are dividing $\left [ 2,3 \right ]$ into four subintervals, set

a = 2, b =3, N = 4 ; the expression becomes

$T_{4}=\frac{1}{8} \left [ f(2 ) +2 f \left (2.25 \right ) +2 f \left (2.5 \right ) + 2f \left ( 2.75 \right ) + f (3 ) \right ]$

where

$f(x)= \log_{3}( x^{2} )$

$f(2)= \log_{3}( 2^{2} ) = \log_{3}4= \frac{\ln 4}{\ln 3} \approx 1.2619$

$f(2.25)= \log_{3}( 2.25^{2} ) = \log_{3}5.0625= \frac{\ln 5.0625}{\ln 3} \approx 1.4763$

$f(2.5)= \log_{3}( 2.5^{2} ) = \log_{3}6.25= \frac{\ln 6.25}{\ln 3} \approx 1.6681$

$f(2.75)= \log_{3}( 2.75^{2} ) = \log_{3}7.5625= \frac{\ln 7.5625}{\ln 3} \approx 1.8416$

$f(3)= \log_{3}( 3^{2} ) = \log_{3}9= 2$

$T_{4}=\frac{1}{8} \left [ 1.2619 +2 (1.4763 )+2 ( 1.6681)+ 2 ( 1.8416) +2 \right]$

$\approx 1.6542$

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Example Question #1114 : Ap Calculus Ab

Solve in the integral by method of trapezoidal sums

$\int_{3}^{5}\cos(e^{3x})dx$

Possible Answers:

$\cos(e^{15})-\cos(e^{9})$

$\cos(e^{5})-\cos(e^{9})$

$\cos(e^{15})+\cos(e^{9})$

$\cos(e^{15})-\sin(e^{9})$

Correct answer:

$\cos(e^{15})-\cos(e^{9})$

Explanation:

To use the trapezoidal rule, we apply the following formula:

$\int_{a}^{b}f(x)dx\approx(b-a)\left [ \frac{f(a)-f(b)}{2}\right ]$

Using the integral from our problem statement, we get

$(5-3)\frac{(\cos(e^{15})-\cos(e^{9}))}{2}=\cos(e^{15})-\cos(e^{9})$

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Example Question #311 : Integrals

Evaluate the integral using the trapezoidal sums method:

$\int_{\frac{\pi}{2}}^{2\pi} e^{sin(x)} dx$

Possible Answers:

$\frac{3\pi}{2}(\frac{1+e}{2})$

$\frac{3\pi}{2}(\frac{2+e}{2})$

$\frac{\pi}{2}(\frac{1+e}{2})$

$\frac{3\pi}{2}(1+e)$

Correct answer:

$\frac{3\pi}{2}(\frac{1+e}{2})$

Explanation:

To solve the integral, we will use the formula for the trapezoidal sums method:

$\int_{a}^{b}f(x)dx=(b-a)\left [ \frac{f(a)+f(b)}{2}\right ]$

Using the integral from the problem statement, we get

$(2\pi-\frac{\pi}{2})\left [ \frac{e^{\sin(2\pi)}+e^{\sin(\frac{\pi}{2})}}{2}\right ]$

Simplifying, we end up with

$\frac{3\pi}{2}(\frac{1+e}{2})$

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Example Question #1115 : Ap Calculus Ab

Evaluate using the trapezoidal approximation:

$\int_{0}^{3}x^2 dx$

Possible Answers:

$\frac{9}{2}$

$\frac{27}{2}$

$\frac{3}{2}$

Correct answer:

$\frac{27}{2}$

Explanation:

The trapezoidal approximation of a definite integral is given by

$\int_{a}^{b}f(x)dx\approx (b-a)(\frac{f(b)+f(a)}{2})$

For our integral, we get

$\int_{0}^{3}x^2 dx \approx (3-0)(\frac{9+0}{2})=\frac{27}{2}$

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Example Question #1116 : Ap Calculus Ab

Use the method of trapezoidal sums to approximate the integral

$\int_{0}^{1}\frac{4x-e^x}{2x}dx$

Possible Answers:

$\frac{3+e}{10}$

$\frac{3-e}{10}$

$\frac{5-e}{10}$

$\frac{3-e}{7}$

Correct answer:

$\frac{3-e}{10}$

Explanation:

To approximate the integral using trapezoidal sums, we use the following formula:

$\int_{a}^{b} f(x)dx=(b-a)\left [ \frac{f(b)+f(a)}{2}\right ]$

Using the integral from the problem statement, we get

$(1-0)\left [ \frac{(\frac{4-e}{5})+(\frac{-1}{5})}{2} \right ]=\frac{3-e}{10}$

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Example Question #1117 : Ap Calculus Ab

Using the method of trapezoidal sums, evaluate the following integral

$\int_{0}^{\frac{\pi}{2}}e^x\sin{x}dx$

Possible Answers:

$\frac{\pi}{2}e^\frac{\pi}{2}$

$\frac{\pi}{4}$

$\frac{\pi}{4}e^\frac{\pi}{2}$

$\frac{\pi}{4}e^\frac{\pi}{4}$

Correct answer:

$\frac{\pi}{4}e^\frac{\pi}{2}$

Explanation:

To find the value of the integral, we use the following formula

$\int_{a}^{b}f(x)dx=\left [ b-a\right ](\frac{f(a)+f(b)}{2})$

$\left [ \frac{\pi}{2}-0 \right ](\frac{\sin(\frac{\pi}{2})e^\frac{\pi}{2}+\sin(0)e^0}{2})=\frac{\pi}{4}e^\frac{\pi}{2}$

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Example Question #1118 : Ap Calculus Ab

Using the method of trapezoidal sums, evaluate the following integral

$\int_{3}^{5}(2xe^{2x+5})dx$

Possible Answers:

$10e^{30}+6e^{14}$

$10e^{30}-6e^{14}$

$5e^{30}-3e^{14}$

$10e^{10}-6e^{3}$

Correct answer:

$10e^{30}+6e^{14}$

Explanation:

To use the method of trapezoidal sums, we follow the definition

$\int_{a}^{b}f(x)dx=(b-a)\left [ \frac{f(a)+f(b)}{2}\right ]$

Using the information from the problem statement, we get

$(5-3)(\frac{10e^{30}+6e^{14}}{2})=10e^{30}+6e^{14}$

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AP Calculus AB : AP Calculus AB

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #1111 : Ap Calculus Ab

Example Question #1112 : Ap Calculus Ab

Example Question #1113 : Ap Calculus Ab

Example Question #41 : Trapezoidal Sums

Example Question #1114 : Ap Calculus Ab

Example Question #311 : Integrals

Example Question #1115 : Ap Calculus Ab

Example Question #1116 : Ap Calculus Ab

Example Question #1117 : Ap Calculus Ab

Example Question #1118 : Ap Calculus Ab

All AP Calculus AB Resources

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