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 #906 : Ap Calculus Ab

$\begin{align*}&\text{The function }f(x)\text{ has the following values on the interval }x=[-8,2]:\\&f(-8)=2\\&f(-6)=-1\\&f(-4)=16\\&f(-2)=12\\&f(0)=10\\&f(2)=-18\\&\text{Approximate the integral of }f(x)\text{ on this interval using left Riemann sums.}\end{align*}$

Possible Answers:

210

Correct answer:

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{In our case, we have function values over an interval:}\\&f(-8)=2\\&f(-6)=-1\\&f(-4)=16\\&f(-2)=12\\&f(0)=10\\&f(2)=-18\\&\text{Although the total interval has a length of }10\\&\text{Notice the points are equidistant, with }5\text{ intervals between.}\\&\text{Each equidistant interval has length: }2\\&\text{Since we are using the left point of each interval, we disregard the larst function value:}\\&\int_{-8}^{2}f(x)\approx2[(2)+(-1)+(16)+(12)+(10)]\\&\int_{-8}^{2}f(x)\approx78\end{align*}$

Report an Error

Example Question #907 : Ap Calculus Ab

$\begin{align*}&\text{The function }f(x)\text{ has the following values on the interval }x=[5,29]:\\&f(5)=8\\&f(13)=-15\\&f(21)=9\\&f(29)=-16\\&\text{Approximate the integral of }f(x)\text{ on this interval using left Riemann sums.}\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{In our case, we have function values over an interval:}\\&f(5)=8\\&f(13)=-15\\&f(21)=9\\&f(29)=-16\\&\text{Although the total interval has a length of }24\\&\text{Notice the points are equidistant, with }3\text{ intervals between.}\\&\text{Each equidistant interval has length: }8\\&\text{Since we are using the left point of each interval, we disregard the larst function value:}\\&\int_{5}^{29}f(x)\approx8[(8)+(-15)+(9)]\\&\int_{5}^{29}f(x)\approx16\end{align*}$

Report an Error

Example Question #201 : Integrals

$\begin{align*}&f(0)=10\\&f(4)=3\\&f(8)=10\\&f(12)=-11\\&f(16)=10\\&\text{Approximate, using a left Riemann sum, }\int_{0}^{16}f(x)\end{align*}$

Possible Answers:

352

Correct answer:

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{In our case, we have function values over an interval:}\\&f(0)=10\\&f(4)=3\\&f(8)=10\\&f(12)=-11\\&f(16)=10\\&\text{Although the total interval has a length of }16\\&\text{Notice the points are equidistant, with }4\text{ intervals between.}\\&\text{Each equidistant interval has length: }4\\&\text{Since we are using the left point of each interval, we disregard the larst function value:}\\&\int_{0}^{16}f(x)\approx4[(10)+(3)+(10)+(-11)]\\&\int_{0}^{16}f(x)\approx48\end{align*}$

Report an Error

Example Question #202 : Integrals

$\begin{align*}&\text{The function }f(x)\text{ has the following values on the interval }x=[-5,5]:\\&f(-5)=-5\\&f(-3)=8\\&f(-1)=4\\&f(1)=12\\&f(3)=-5\\&f(5)=-12\\&\text{Approximate the integral of }f(x)\text{ on this interval using left Riemann sums.}\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{In our case, we have function values over an interval:}\\&f(-5)=-5\\&f(-3)=8\\&f(-1)=4\\&f(1)=12\\&f(3)=-5\\&f(5)=-12\\&\text{Although the total interval has a length of }10\\&\text{Notice the points are equidistant, with }5\text{ intervals between.}\\&\text{Each equidistant interval has length: }2\\&\text{Since we are using the left point of each interval, we disregard the larst function value:}\\&\int_{-5}^{5}f(x)\approx2[(-5)+(8)+(4)+(12)+(-5)]\\&\int_{-5}^{5}f(x)\approx28\end{align*}$

Report an Error

Example Question #21 : Riemann Sums (Left, Right, And Midpoint Evaluation Points)

$\begin{align*}&\text{The function }f(x)\text{ has the following values on the interval }x=[-4,11]:\\&f(-4)=2\\&f(-1)=-11\\&f(2)=6\\&f(5)=-1\\&f(8)=-14\\&f(11)=12\\&\text{Approximate the integral of }f(x)\text{ on this interval using left Riemann sums.}\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{In our case, we have function values over an interval:}\\&f(-4)=2\\&f(-1)=-11\\&f(2)=6\\&f(5)=-1\\&f(8)=-14\\&f(11)=12\\&\text{Although the total interval has a length of }15\\&\text{Notice the points are equidistant, with }5\text{ intervals between.}\\&\text{Each equidistant interval has length: }3\\&\text{Since we are using the left point of each interval, we disregard the larst function value:}\\&\int_{-4}^{11}f(x)\approx3[(2)+(-11)+(6)+(-1)+(-14)]\\&\int_{-4}^{11}f(x)\approx-54\end{align*}$

Report an Error

Example Question #22 : Riemann Sums (Left, Right, And Midpoint Evaluation Points)

$\begin{align*}&f(-2)=-17\\&f(1)=-4\\&f(4)=-16\\&f(7)=-16\\&\text{Approximate, using a left Riemann sum, }\int_{-2}^{7}f(x)\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{In our case, we have function values over an interval:}\\&f(-2)=-17\\&f(1)=-4\\&f(4)=-16\\&f(7)=-16\\&\text{Although the total interval has a length of }9\\&\text{Notice the points are equidistant, with }3\text{ intervals between.}\\&\text{Each equidistant interval has length: }3\\&\text{Since we are using the left point of each interval, we disregard the larst function value:}\\&\int_{-2}^{7}f(x)\approx3[(-17)+(-4)+(-16)]\\&\int_{-2}^{7}f(x)\approx-111\end{align*}$

Report an Error

Example Question #23 : Riemann Sums (Left, Right, And Midpoint Evaluation Points)

$\begin{align*}&\text{Approximate }\int_{5}^{23}f(x)\\&\text{Using a left Riemann sum, if }f(x)\text{ has the values:}\\&f(5)=-3\\&f(11)=8\\&f(17)=11\\&f(23)=-3\end{align*}$

Possible Answers:

234

Correct answer:

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{In our case, we have function values over an interval:}\\&f(5)=-3\\&f(11)=8\\&f(17)=11\\&f(23)=-3\\&\text{Although the total interval has a length of }18\\&\text{Notice the points are equidistant, with }3\text{ intervals between.}\\&\text{Each equidistant interval has length: }6\\&\text{Since we are using the left point of each interval, we disregard the larst function value:}\\&\int_{5}^{23}f(x)\approx6[(-3)+(8)+(11)]\\&\int_{5}^{23}f(x)\approx96\end{align*}$

Report an Error

Example Question #24 : Riemann Sums (Left, Right, And Midpoint Evaluation Points)

$\begin{align*}&\text{Approximate }\int_{-8}^{19}f(x)\\&\text{Using a left Riemann sum, if }f(x)\text{ has the values:}\\&f(-8)=-10\\&f(1)=12\\&f(10)=-1\\&f(19)=11\end{align*}$

Possible Answers:

324

108

Correct answer:

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{In our case, we have function values over an interval:}\\&f(-8)=-10\\&f(1)=12\\&f(10)=-1\\&f(19)=11\\&\text{Although the total interval has a length of }27\\&\text{Notice the points are equidistant, with }3\text{ intervals between.}\\&\text{Each equidistant interval has length: }9\\&\text{Since we are using the left point of each interval, we disregard the larst function value:}\\&\int_{-8}^{19}f(x)\approx9[(-10)+(12)+(-1)]\\&\int_{-8}^{19}f(x)\approx9\end{align*}$

Report an Error

Example Question #21 : Riemann Sums (Left, Right, And Midpoint Evaluation Points)

Let $f(x) = \frac{1}{7}x^{7} - \frac{6}{5} x^{5} + \frac{5}{3}x^{3} - 30x$ .

A relative maximum of the graph of can be located at:

Possible Answers:

The graph of has no relative maximum.

$x=- \sqrt[4]{5}$

$x= \sqrt[4]{5}$

$x =- \sqrt {6}$

$x = \sqrt {6}$

Correct answer:

$x =- \sqrt {6}$

Explanation:

At a relative minimum (c,f(c)) of the graph f(x) , it will hold that f'(c) = 0 and f''(c) < 0 .

First, find f'(x) . Using the sum rule,

$f'(x) = \frac{\mathrm{d} }{\mathrm{d} x}\left ( \frac{1}{7}x^{7} - \frac{6}{5} x^{5} + \frac{5}{3}x^{3} - 30x \right )$

$f'(x) = \frac{\mathrm{d} }{\mathrm{d} x}\left ( \frac{1}{7}x^{7} \right )- \frac{\mathrm{d} }{\mathrm{d} x}\left ( \frac{6}{5} x^{5} \right )+ \frac{\mathrm{d} }{\mathrm{d} x}\left ( \frac{5}{3}x^{3} \right )- \frac{\mathrm{d} }{\mathrm{d} x}\left ( 30x \right )$

Differentiate the individual terms using the Constant Multiple and Power Rules:

$f'(x) = \frac{1}{7} \cdot 7 x^{6} - \frac{6}{5} \cdot 5 x^{4} + \frac{5}{3} \cdot 3 x^{2} - 30x$

$f'(x) = x^{6} - 6 x^{4} +5 x^{2} - 30$

Set this equal to 0:

$x^{6} - 6 x^{4} +5 x^{2} - 30 = 0$

$(x^{6} - 6 x^{4} )+(5 x^{2} - 30) = 0$

$x^{4} (x^{2} - 6 )+ 5 (x^{2} - 6 ) = 0$

$(x^{4}+ 5 ) (x^{2} - 6 ) = 0$

Either:

$x^{4}+5 = 0$ , in which case, $x^{4} = -5$ ; this equation has no real solutions.

$x^{2}- 6=0$ has two real solutions, $-\sqrt {6}$ and $\sqrt {6}$ .

Now take the second derivative, again using the sum rule:

$f'(x) = x^{6} - 6 x^{4} +5 x^{2} - 30$

$f''(x) = \frac{\mathrm{d} }{\mathrm{d} x}(x^{6} - 6 x^{4} +5 x^{2} - 30)$

$f''(x) = \frac{\mathrm{d} }{\mathrm{d} x}(x^{6} )- \frac{\mathrm{d} }{\mathrm{d} x} ( 6 x^{4} )+\frac{\mathrm{d} }{\mathrm{d} x}( 5 x^{2} )- \frac{\mathrm{d} }{\mathrm{d} x}( 30)$

Differentiate the individual terms using the Constant Multiple and Power Rules:

$f''(x) =6 x^{5} - 6 ( 4 x^{3} )+ 5( 2x )- 0$

$f''(x) =6 x^{5} -2 4 x^{3} +10x$

Substitute $\sqrt {6}$ for :

$f''( \sqrt {6}) =6 ( \sqrt {6} )^{5} -2 4 ( \sqrt {6})^{3} +10( \sqrt {6})$

$f''(-\sqrt {6}) = 216\sqrt {6}-144 \sqrt {6} +10\sqrt {6}$

$f''(-\sqrt {6}) =82\sqrt {6} >0$

Therefore, has a relative minimum at $x = \sqrt {6}$ .

Now. substitute $-\sqrt {6}$ for :

$f''(-\sqrt {6}) =6 (-\sqrt {6} )^{5} -2 4 (-\sqrt {6})^{3} +10(-\sqrt {6})$

$f''(-\sqrt {6}) =- 216\sqrt {6}+144 \sqrt {6} -10\sqrt {6}$

$f''(-\sqrt {6}) =-82\sqrt {6} < 0$

Therefore, has a relative maximum at $x =- \sqrt {6}$ .

Report an Error

Example Question #911 : Ap Calculus Ab

Estimate the integral of f(x)=3x^2+1 from 0 to 3 using left-Riemann sum and 6 rectangles. Use $\Delta x=0.5$

Possible Answers:

74.25

23.125

46.25

23.625

37.125

Correct answer:

23.625

Explanation:

Because our $\Delta x$ is constant, the left Riemann sum will be

$R_{L}=\Delta x(f(0)+f(0.5)+f(1)+f(1.5)+f(2)+f(2.5))$

$R_{L}=0.5\cdot (1+1.75+4+7.75+13+19.75)=23.625$

Report an Error

View AP Calculus AB Tutors

Nevryk
Certified Tutor

University of Nevada-Las Vegas, Bachelor of Science, Computer Science.

View AP Calculus AB Tutors

Donald
Certified Tutor

University of Georgia, Bachelors, Latin.

View AP Calculus AB Tutors

Huanzhi Isabel
Certified Tutor

Texas A & M University-College Station, Bachelor of Science, Biochemistry.

All AP Calculus AB Resources

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

Popular Subjects

Reading Tutors in Seattle, ISEE Tutors in New York City, GMAT Tutors in Chicago, ISEE Tutors in Atlanta, French Tutors in New York City, Calculus Tutors in Boston, ACT Tutors in Seattle, ISEE Tutors in San Francisco-Bay Area, ACT Tutors in Dallas Fort Worth, Reading Tutors in Dallas Fort Worth

Popular Courses & Classes

ISEE Courses & Classes in Dallas Fort Worth, LSAT Courses & Classes in Denver, SAT Courses & Classes in San Diego, GMAT Courses & Classes in Boston, SSAT Courses & Classes in New York City, GMAT Courses & Classes in Philadelphia, MCAT Courses & Classes in Washington DC, Spanish Courses & Classes in Los Angeles, SAT Courses & Classes in Chicago, Spanish Courses & Classes in New York City

Popular Test Prep

MCAT Test Prep in Atlanta, SAT Test Prep in Washington DC, SAT Test Prep in Philadelphia, SSAT Test Prep in San Diego, MCAT Test Prep in Boston, ISEE Test Prep in Dallas Fort Worth, SSAT Test Prep in Dallas Fort Worth, GMAT Test Prep in Atlanta, ACT Test Prep in Miami, MCAT Test Prep in Houston

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 #906 : Ap Calculus Ab

Example Question #907 : Ap Calculus Ab

Example Question #201 : Integrals

Example Question #202 : Integrals

Example Question #21 : Riemann Sums (Left, Right, And Midpoint Evaluation Points)

Example Question #22 : Riemann Sums (Left, Right, And Midpoint Evaluation Points)

Example Question #23 : Riemann Sums (Left, Right, And Midpoint Evaluation Points)

Example Question #24 : Riemann Sums (Left, Right, And Midpoint Evaluation Points)

Example Question #21 : Riemann Sums (Left, Right, And Midpoint Evaluation Points)

Example Question #911 : Ap Calculus Ab

All AP Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: