Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 1 : Midpoint Riemann Sums

Study concepts, example questions & explanations for Calculus 1

Create An Account

All Calculus 1 Resources

10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #91 : How To Find Midpoint Riemann Sums

Using the method of midpoint Riemann sums, approximate the integral $\int_0^9 1.2^{x+1}$ using three midpoints.

Possible Answers:

9.01

15.13

31.90

22.16

27.04

Correct answer:

27.04

Explanation:

A Riemann sum integral approximation over an interval [a,b] with subintervals follows the form:

$\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))$

It is essentially a sum of rectangles each with a base of length $\frac{b-a}{n}$ and variable heights f(x_i) , which depend on the function value at a given point x_i .

We are approximating $\int_0^9 1.2^{x+1}$

So the interval is [0,9] , the subintervals have length $\frac{9-0}{3}=3$ , and since we are using the midpoints of each interval, the x-values are [1.5,4.5,7.5]

$\int_0^9 1.2^{x+1}\approx 3[1.2^{1.5+1}+1.2^{4.5+1}+1.2^{7.5+1}]$

$\int_0^9 1.2^{x+1}\approx 27.04$

Report an Error

Example Question #92 : How To Find Midpoint Riemann Sums

Approximate the integral $\int_3^7 1.02^{x^3}$ using the method of midpoint Riemann sums and four midpoints.

Possible Answers:

11.2

23.7

8.9

265.4

39.2

Correct answer:

265.4

Explanation:

A Riemann sum integral approximation over an interval [a,b] with subintervals follows the form:

$\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))$

It is essentially a sum of rectangles each with a base of length $\frac{b-a}{n}$ and variable heights f(x_i) , which depend on the function value at a given point x_i .

We are approximating $\int_3^7 1.02^{x^3}$

So the interval is [3,7] , the subintervals have length $\frac{7-3}{4}=1$ , and since we are using the midpoints of each interval, the x-values are [3.5,4.5,5.5,6.5]

$\int_3^7 1.02^{x^3}\approx 1[1.02^{3.5^3}+1.02^{4.5^3}+1.02^{5.5^3}+1.02^{6.5^3}]$

$\int_3^7 1.02^{x^3} \approx 265.4$

Report an Error

Example Question #93 : How To Find Midpoint Riemann Sums

Using the method of midpoint Riemann sums, approximate the integral $\int_{-1.2}^{2.4}cos(1.7^{x^{1.3}})$ using four midpoints.

Possible Answers:

0.149+0.115i

0.149-0.115i

0.115-0.149i

0.231+i

0.115+0.149i

Correct answer:

0.115+0.149i

Explanation:

A Riemann sum integral approximation over an interval [a,b] with subintervals follows the form:

$\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))$

It is essentially a sum of rectangles each with a base of length $\frac{b-a}{n}$ and variable heights f(x_i) , which depend on the function value at a given point x_i .

We're approximating $\int_{-1.2}^{2.4}cos(1.7^{x^{1.3}})$

So the interval is [-1.2,2.4] , the subintervals have length $\frac{2.4-(-1.2)}{4}=0.9$ , and since we are using the midpoints of each interval, the x-values are [-0.75,0.15,1.05,1.95]

${\int_{-1.2}^{2.4}cos(1.7^{x^{1.3}})\approx 0.9[cos(1.7^{(-0.75)^{1.3}})+cos(1.7^{(0.15)^{1.3}})+cos(1.7^{(1.05)^{1.3}})+cos(1.7^{(1.95)^{1.3}})]}$

$\int_{-1.2}^{2.4}cos(1.7^{x^{1.3}}) \approx 0.115+0.149i$

Report an Error

Example Question #91 : How To Find Midpoint Riemann Sums

Using the method of midpoint of Reimann sums, approximate the integral $\int_0^3 tan(x^2)dx$ using three midpoints.

Possible Answers:

-0.81

2.31

0.49

-1.02

0.78

Correct answer:

-1.02

Explanation:

A Reimann sum integral approximation over an interval [a,b] with subintervals follows the form:

$\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))$

It is essentially a sum of rectangles each with a base of length $\frac{b-a}{n}$ and variable heights f(x_i) , which depend on the function value at a given point x_i .

We're approximating $\int_0^3 tan(x^2)dx$

So the interval is [0,3] , the subintervals have length $\frac{3-0}{3}=1$ , and since we are using the midpoints of each interval, the x-values are [0.5,1.5,2.5]

$\int_0^3 tan(x^2)dx \approx 1[ tan(0.5^2)+ tan(1.5^2)+ tan(2.5^2)]=-1.02$

Report an Error

Example Question #94 : How To Find Midpoint Riemann Sums

Using the method of midpoint Reimann sums, approximate the integral $\int_1^5 ln(tan(2^x))dx$ using four midpoints.

Possible Answers:

-0.64 +9.42i

9.42-0.64 i

-0.64 -9.42i

9.42+0.64 i

-9.42-0.64 i

Correct answer:

-0.64 +9.42i

Explanation:

A Reimann sum integral approximation over an interval [a,b] with subintervals follows the form:

$\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))$

It is essentially a sum of rectangles each with a base of length $\frac{b-a}{n}$ and variable heights f(x_i) , which depend on the function value at a given point x_i .

We're approximating $\int_1^5 ln(tan(2^x))dx$

So the interval is [1,5] , the subintervals have length $\frac{5-1}{4}=1$ , and since we are using the midpoints of each interval, the x-values are [1.5,2.5,3.5,4.5]

$\int_1^5 ln(tan(2^x))dx \approx 1[ln(tan(2^{1.5}))+ln(tan(2^{2.5}))+ln(tan(2^{3.5}))+ln(tan(2^{4.5}))]$

$\int_1^5 ln(tan(2^x))dx \approx -0.64 +9.42i$

Report an Error

Example Question #91 : Functions

Using the method of midpoint Reimann sums, approximate the integral $\int_2^8 \frac{1}{tan(3^x)}dx$ using two midpoints.

Possible Answers:

6.1

-19.5

-6.5

4.5

18.3

Correct answer:

-19.5

Explanation:

A Reimann sum integral approximation over an interval [a,b] with subintervals follows the form:

$\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))$

It is essentially a sum of rectangles each with a base of length $\frac{b-a}{n}$ and variable heights f(x_i) , which depend on the function value at a given point x_i .

We're approximating $\int_2^8 \frac{1}{tan(3^x)}dx$

So the interval is [2,8] , the subintervals have length $\frac{8-2}{2}=3$ , and since we are using the midpoints of each interval, the x-values are [3.5,6.5]

$\int_2^8 \frac{1}{tan(3^x)}dx \approx 3[\frac{1}{tan(3^{3.5})}+\frac{1}{tan(3^{6.5})}]=-19.5$

Report an Error

Example Question #96 : How To Find Midpoint Riemann Sums

Using the method of midpoint Reimann sums, approximate the integral $\int_3^9 \frac{3^{x}}{2^{x}}dx$ using three midpoints.

Possible Answers:

28.1

42.1

192.6

96.3

84.2

Correct answer:

84.2

Explanation:

A Reimann sum integral approximation over an interval [a,b] with subintervals follows the form:

$\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))$

It is essentially a sum of rectangles each with a base of length $\frac{b-a}{n}$ and variable heights f(x_i) , which depend on the function value at a given point x_i .

We're approximating $\int_3^9 \frac{3^{x}}{2^{x}}dx$

So the interval is [3,9] , the subintervals have length $\frac{9-3}{3}=2$ , and since we are using the midpoints of each interval, the x-values are [4,6,8]

$\int_3^9 \frac{3^{x}}{2^{x}}dx \approx 2[\frac{3^{4}}{2^{4}}+\frac{3^{6}}{2^{6}}+\frac{3^{8}}{2^{8}}]=84.2$

Report an Error

Example Question #97 : How To Find Midpoint Riemann Sums

Using the method of midpoint Reimann sums, approximate the integral $\int_1^{2.5} tan(\frac{1}{x^{1.4}})$ using three midpoints.

Possible Answers:

0.348

0.861

0.696

1.722

0.132

Correct answer:

0.861

Explanation:

A Reimann sum integral approximation over an interval [a,b] with subintervals follows the form:

$\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))$

It is essentially a sum of rectangles each with a base of length $\frac{b-a}{n}$ and variable heights f(x_i) , which depend on the function value at a given point x_i .

We're approximating $\int_1^{2.5} tan(\frac{1}{x^{1.4}})$

So the interval is [1,2.5] , the subintervals have length $\frac{2.5-1}{3}=0.5$ , and since we are using the midpoints of each interval, the x-values are [1.25,1.75,2.25]

$\int_1^{2.5} tan(\frac{1}{x^{1.4}})\approx 0.5[tan(\frac{1}{1.25^{1.4}})+tan(\frac{1}{1.75^{1.4}})+tan(\frac{1}{2.25^{1.4}})]$

$\int_1^{2.5} tan(\frac{1}{x^{1.4}})\approx 0.861$

Report an Error

Example Question #91 : Differential Functions

Using the method of midpoint Reimann sums, approximate the integral $\int_{2.5}^{4.5} \frac{cos(x^3)}{tan(x^4)}dx$ using four midpoints.

Possible Answers:

1.904

0.952

3.312

1.656

3.808

Correct answer:

1.904

Explanation:

A Reimann sum integral approximation over an interval [a,b] with subintervals follows the form:

$\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))$

It is essentially a sum of rectangles each with a base of length $\frac{b-a}{n}$ and variable heights f(x_i) , which depend on the function value at a given point x_i .

We're approximating $\int_{2.5}^{4.5} \frac{cos(x^3)}{tan(x^4)}dx$

So the interval is [2.5,4.5] , the subintervals have length $\frac{4.5-2.5}{4}=0.5$ , and since we are using the midpoints of each interval, the x-values are [2.75,3.25,3.75,4.25]

$\int_{2.5}^{4.5} \frac{cos(x^3)}{tan(x^4)}dx \approx 0.5[\frac{cos(2.75^3)}{tan(2.75^4)}+\frac{cos(3.25^3)}{tan(3.25^4)}+\frac{cos(3.75^3)}{tan(3.75^4)}+\frac{cos(4.25^3)}{tan(4.25^4)}]$

$\int_{2.5}^{4.5} \frac{cos(x^3)}{tan(x^4)}dx \approx 1.904$

Report an Error

Example Question #99 : How To Find Midpoint Riemann Sums

Using the method of midpoint Reimann sums, approximate the integral $\int_{10.5}^{13.5} tan(e^{x^{1.5}})dx$ using three midpoints.

Possible Answers:

0.229

0.782

-0.014

1.114

-1.083

Correct answer:

0.782

Explanation:

A Reimann sum integral approximation over an interval [a,b] with subintervals follows the form:

$\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))$

It is essentially a sum of rectangles each with a base of length $\frac{b-a}{n}$ and variable heights f(x_i) , which depend on the function value at a given point x_i .

We're approximating $\int_{10.5}^{13.5} tan(e^{x^{1.5}})dx$

So the interval is [10.5,13.5] , the subintervals have length $\frac{13.5-10.5}{3}=1$ , and since we are using the midpoints of each interval, the x-values are [11,12,13]

$\int_{10.5}^{13.5} tan(e^{x^{1.5}})dx \approx 1[tan(e^{11^{1.5}})+tan(e^{12^{1.5}})+tan(e^{13^{1.5}})]$

$\int_{10.5}^{13.5} tan(e^{x^{1.5}})dx \approx 0.782$

Report an Error

1743693617 mini magick20250403 191 1wig238

View Calculus Tutors

Hayden
Certified Tutor

University of Arizona, Bachelor of Science, Systems Engineering.

View Calculus Tutors

Mohammad
Certified Tutor

Jamia Millia Islamia, Master of Science, Physics.

View Calculus Tutors

Rachna
Certified Tutor

The Texas A&M University System Office, Bachelor of Science, Applied Mathematics.

All Calculus 1 Resources

10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Computer Science Tutors in Philadelphia, Reading Tutors in Phoenix, GMAT Tutors in San Francisco-Bay Area, GRE Tutors in Miami, Math Tutors in Chicago, Chemistry Tutors in Houston, GMAT Tutors in Atlanta, French Tutors in Dallas Fort Worth, Chemistry Tutors in Chicago, Statistics Tutors in Miami

Popular Courses & Classes

ISEE Courses & Classes in Boston, SSAT Courses & Classes in Boston, SSAT Courses & Classes in Seattle, GMAT Courses & Classes in Dallas Fort Worth, LSAT Courses & Classes in Seattle, GMAT Courses & Classes in San Francisco-Bay Area, GMAT Courses & Classes in Seattle, LSAT Courses & Classes in Boston, ISEE Courses & Classes in Denver, SAT Courses & Classes in Atlanta

Popular Test Prep

ISEE Test Prep in San Francisco-Bay Area, LSAT Test Prep in Miami, ISEE Test Prep in Philadelphia, ISEE Test Prep in Los Angeles, ISEE Test Prep in Boston, ISEE Test Prep in San Diego, LSAT Test Prep in Chicago, MCAT Test Prep in Philadelphia, MCAT Test Prep in Los Angeles, SAT Test Prep in Seattle

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

Calculus 1 : Midpoint Riemann Sums

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #91 : How To Find Midpoint Riemann Sums

Example Question #92 : How To Find Midpoint Riemann Sums

Example Question #93 : How To Find Midpoint Riemann Sums

Example Question #91 : How To Find Midpoint Riemann Sums

Example Question #94 : How To Find Midpoint Riemann Sums

Example Question #91 : Functions

Example Question #96 : How To Find Midpoint Riemann Sums

Example Question #97 : How To Find Midpoint Riemann Sums

Example Question #91 : Differential Functions

Example Question #99 : How To Find Midpoint Riemann Sums

All Calculus 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: