We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

AP Calculus AB : Numerical approximations to definite integrals

Study concepts, example questions & explanations for AP Calculus AB

Create An Account

All AP Calculus AB Resources

3 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #33 : Trapezoidal Sums

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

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

Report an Error

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

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

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

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

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

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

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

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

$\approx 1.6542$

Report an Error

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

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

Report an Error

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

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

Report an Error

Example Question #41 : Trapezoidal Sums

Evaluate using the trapezoidal approximation:

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

Possible Answers:

$\frac{3}{2}$

$\frac{27}{2}$

$\frac{9}{2}$

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

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

Report an Error

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

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

Report an Error

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

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

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

Report an Error

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

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

Report an Error

Example Question #316 : Integrals

Use the trapezoidal approximation to solve the definite integral, and find the difference between it and the actual integral:

$\int_{0}^{5} x^4 dx$

Possible Answers:

$\frac{3075}{2}$

$\frac{1875}{2}$

$\frac{3125}{2}$

Correct answer:

$\frac{1875}{2}$

Explanation:

The trapezoidal approximation to definite integrals is given by

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

Using this formula for our integral, we get

$\int_{0}^{5} x^4 dx \approx (5-0)(\frac{625-0}{2})=\frac{3125}{2}$

Actually integrating, we get

$\int_{0}^{5} x^4 dx = \frac{x^5}{5}\left.\begin{matrix} 5\\0 \end{matrix}\right| = \frac{5^5}{5}=5^4=625$

The rule used for integration is

$\int x^n dx=\frac{x^{n+1}}{n+1}+C$

The difference between the approximation and the actual answer is

$\frac{3125}{2}-625=\frac{1875}{2}$

Report an Error

Example Question #316 : Integrals

Approximate the value of $\int_{-2}^{4} x^2 dx$ using a trapezoidal sum with step size h = 2 . How far away is this approximation from the actual value of the integral above?

Possible Answers:

Correct answer:

Explanation:

Trapezoidal sums are found by creating trapezoids whose left and right end points are on the specified function, and whose widths are the step size. We then sum up their areas by remembering that the area of a trapezoid is the base times the average of the heights.

Thus, the calculation of the trapezoidal sum for this example would be $2 \cdot [\frac{f(-2)+f(0)}{2}] + 2 \cdot [\frac{f(0)+f(2)}{2}] + 2 \cdot [\frac{f(2)+f(4)}{2}]$

A more simplified version would be given by f(-2) + 2(f(0) + f(2)) + f(4)

Which evaluates to

4 + 2(0 + 4) + 16 = 28

The actual answer is found by evaluating the definite integral given, which would just be given by $\int_{-2}^{4} x^2 dx = \left . \frac{x^3}{3} \right ]^4_{-2} = \frac{4^3 - (-2)^3}{3} = \frac{64 + 8}{3} = \frac{72}{3} = 24$

The difference between the approximation and the the true answer is thus 28-24 = 4

Report an Error

View AP Calculus AB Tutors

Dalton
Certified Tutor

University of Massachusetts Amherst, Doctor of Philosophy, Mathematics.

View AP Calculus AB Tutors

Vandana
Certified Tutor

Jawaharlal Nehru Technological University, Bachelor of Technology, Civil Engineering. Florida International University, Maste...

View AP Calculus AB Tutors

Satweka
Certified Tutor

The University of Texas at Dallas, Master of Science, Biomedical Engineering.

All AP Calculus AB Resources

3 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Statistics Tutors in Seattle, Algebra Tutors in Denver, ACT Tutors in Denver, Chemistry Tutors in Phoenix, MCAT Tutors in Chicago, Chemistry Tutors in Seattle, Statistics Tutors in San Francisco-Bay Area, ACT Tutors in Miami, Biology Tutors in Seattle, Statistics Tutors in Los Angeles

Popular Courses & Classes

ISEE Courses & Classes in Washington DC, SAT Courses & Classes in San Francisco-Bay Area, MCAT Courses & Classes in Dallas Fort Worth, GMAT Courses & Classes in Dallas Fort Worth, SAT Courses & Classes in Boston, SAT Courses & Classes in Washington DC, Spanish Courses & Classes in Phoenix, ACT Courses & Classes in Boston, MCAT Courses & Classes in Phoenix, LSAT Courses & Classes in San Diego

Popular Test Prep

ISEE Test Prep in Los Angeles, ACT Test Prep in Boston, SAT Test Prep in Philadelphia, GRE Test Prep in Los Angeles, LSAT Test Prep in Philadelphia, LSAT Test Prep in San Diego, ACT Test Prep in San Francisco-Bay Area, SSAT Test Prep in Seattle, ACT Test Prep in Dallas Fort Worth, SAT Test Prep in Dallas Fort Worth

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

AP Calculus AB : Numerical approximations to definite integrals

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #33 : Trapezoidal Sums

Example Question #41 : Trapezoidal Sums

Example Question #1114 : Ap Calculus Ab

Example Question #311 : Integrals

Example Question #41 : Trapezoidal Sums

Example Question #1116 : Ap Calculus Ab

Example Question #1117 : Ap Calculus Ab

Example Question #1118 : Ap Calculus Ab

Example Question #316 : Integrals

Example Question #316 : Integrals

All AP Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: