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 #1151 : Calculus

Utilize the method of midpoint Riemann sums to approximate $\int_{0.9}^{0.96}cos(4^{tan(x)})dx$ using three midpoints.

Possible Answers:

2.107

-0.023

0.054

-1.008

-0.349

Correct answer:

0.054

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 asked to approximate $\int_{0.9}^{0.96}cos(4^{tan(x)})dx$

So the interval is [0.9,0.96] , the subintervals have length $\frac{0.96-0.9}{3}=0.02$ , and since we are using the midpoints of each interval, the x-values are [0.91,0.93,0.95]

$\int_{0.9}^{0.96}cos(4^{tan(x)})dx\approx (0.02)[cos(4^{tan(0.91)})+cos(4^{tan(0.93)})+cos(4^{tan(0.95)})]$

$\int_{0.9}^{0.96}cos(4^{tan(x)})dx\approx 0.054$

Report an Error

Example Question #132 : Midpoint Riemann Sums

Utilize the method of midpoint Riemann sums to approximate $\int_{3.5}^{4.3}(cos(x))^{cos(x)}dx$ using four midpoints.

Possible Answers:

0.513+0.735i

-0.513+0.735i

-0.513-0.735i

0.513-0.735i

Correct answer:

-0.513+0.735i

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 asked to approximate $\int_{3.5}^{4.3}(cos(x))^{cos(x)}dx$

So the interval is [3.5,4.3] , the subintervals have length $\frac{4.3-3.5}{4}=0.2$ , and since we are using the midpoints of each interval, the x-values are [3.6,3.8,4.0,4.2]

$\int_{3.5}^{4.3}(cos(x))^{cos(x)}dx\approx (0.2)[(cos(3.6))^{cos(3.6)}+(cos(3.8))^{cos(3.8)}+(cos(4.0))^{cos(4.0)}+(cos(4.2))^{cos(4.2)}]$

$\int_{3.5}^{4.3}(cos(x))^{cos(x)}dx\approx-0.513+0.735i$

Report an Error

Example Question #133 : Midpoint Riemann Sums

Utilize the method of midpoint Riemann sums to approximate $\int_{0.2}^{1.4}(sin(x))^{\frac{1}{cos(x)}}dx$ using three midpoints.

Possible Answers:

0.476

1.398

0.098

2.101

0.721

Correct answer:

0.721

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 asked to approximate $\int_{0.2}^{1.4}(sin(x))^{\frac{1}{cos(x)}}dx$

So the interval is [0.2,1.4] , the subintervals have length $\frac{1.4-0.2}{3}=0.4$ , and since we are using the midpoints of each interval, the x-values are [0.4,0.8,1.2]

$\int_{0.2}^{1.4}(sin(x))^{\frac{1}{cos(x)}}dx\approx (0.4)[(sin(0.4))^{\frac{1}{cos(0.4)}}+(sin(0.8))^{\frac{1}{cos(0.8)}}+(sin(1.2))^{\frac{1}{cos(1.2)}}]$

$\int_{0.2}^{1.4}(sin(x))^{\frac{1}{cos(x)}}dx\approx 0.721$

Report an Error

Example Question #134 : Midpoint Riemann Sums

Utilize the method of midpoint Riemann sums to approximate $\int_{2.6}^{3.2}2.7^{sin(x^2)}dx$ using three midpoints.

Possible Answers:

4.454

1.095

6.138

0.893

2.235

Correct answer:

1.095

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 asked to approximate $\int_{2.6}^{3.2}2.7^{sin(x^2)}dx$

So the interval is [2.6,3.2] , the subintervals have length $\frac{3.6-2.6}{3}=0.2$ , and since we are using the midpoints of each interval, the x-values are [2.7,2.9,3.1]

$\int_{2.6}^{3.2}2.7^{sin(x^2)}dx\approx (0.2)[2.7^{sin(2.7^2)}+2.7^{sin(2.9^2)}+2.7^{sin(3.1^2)}]$

$\int_{2.6}^{3.2}2.7^{sin(x^2)}dx\approx 1.095$

Report an Error

Example Question #135 : Midpoint Riemann Sums

Utilize the method of midpoint Riemann sums to approximate $\int_{0.5}^{2.9}(sin(x^{tan(x^{cos(x)})}))dx$ using two midpoints.

Possible Answers:

0.647

2.898

2.296

3.121

1.961

Correct answer:

2.296

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 asked to approximate $\int_{0.5}^{2.9}(sin(x^{tan(x^{cos(x)})}))dx$

So the interval is [0.5,2.9] , the subintervals have length $\frac{2.9-0.5}{2}=1.2$ , and since we are using the midpoints of each interval, the x-values are [1.1,2.3]

$\int_{0.5}^{2.9}(sin(x^{tan(x^{cos(x)})}))\approx (1.2)[(sin(1.1^{tan(1.1^{cos(1.1)})}))dx+(sin(2.3^{tan(2.3^{cos(2.3)})}))]$

$\int_{0.5}^{2.9}(sin(x^{tan(x^{cos(x)})}))dx\approx 2.296$

Report an Error

Example Question #136 : Midpoint Riemann Sums

Utilize the method of midpoint Riemann sums to approximate $\int_{1}^{100} tan(\sqrt[3]{x})dx$ using three midpoints.

Possible Answers:

52.418

77.277

93.248

64.009

104.522

Correct answer:

93.248

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 asked to approximate $\int_{1}^{100} tan(\sqrt[3]{x})dx$

So the interval is [1,100] , the subintervals have length $\frac{100-1}{3}=33$ , and since we are using the midpoints of each interval, the x-values are [17.5,50.5,83.5]

$\int_{1}^{100} tan(\sqrt[3]{x})dx\approx (33)[tan(\sqrt[3]{17.5})+tan(\sqrt[3]{50.5})+tan(\sqrt[3]{83.5})]$

$\int_{1}^{100} tan(\sqrt[3]{x})dx\approx93.248$

Report an Error

Example Question #137 : Midpoint Riemann Sums

Utilize the method of midpoint Riemann sums to approximate $\int_{0.0003}^{0.0006}14^{cos(\frac{1}{x})}dx$ using three midpoints.

Possible Answers:

0.02004

0.00012

0.00078

1.00089

0.39765

Correct answer:

0.00012

Explanation:

A Riemann sum integral approximation over an interval [a,b] with subintervals follo $\int_{0.0003}^{0.0006}14^{cos(\frac{1}{x})}dx\approx 0.00012$ ws 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 asked to approximate $\int_{0.0003}^{0.0006}14^{cos(\frac{1}{x})}dx$

So the interval is [0.0003,0.0006] , the subintervals have length $\frac{0.0006-0.0003}{3}=0.0001$ , and since we are using the midpoints of each interval, the x-values are [0.00035,0.00045,0.00055]

$\int_{0.0003}^{0.0006}14^{cos(\frac{1}{x})}dx\approx (0.0001)[14^{cos(\frac{1}{0.00035})}+14^{cos(\frac{1}{0.00045})}+14^{cos(\frac{1}{0.00055})}]$

Report an Error

Example Question #138 : Midpoint Riemann Sums

Utilize the method of midpoint Riemann sums to approximate $\int_{1}^{7}tan(cos(x^2+x))dx$ using three midpoints.

Possible Answers:

10.080

2.877

16.055

14.932

7.173

Correct answer:

2.877

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 asked to approximate $\int_{1}^{7}tan(cos(x^2+x))dx$

So the interval is [1,7] , the subintervals have length $\frac{7-1}{3}=2$ , and since we are using the midpoints of each interval, the x-values are [2,4,6]

${\int_{1}^{7}tan(cos(x^2+x))dx\approx (2)[tan(cos(2^2+2))+tan(cos(4^2+4))+tan(cos(6^2+6))]}$

$\int_{1}^{7}tan(cos(x^2+x))dx\approx 2.877$

Report an Error

Example Question #139 : Midpoint Riemann Sums

Utilize the method of midpoint Riemann sums to approximate $\int_{3}^{15}x!dx$ using three midpoints.

Possible Answers:

159687384

87181920720

6227383800

348727682880

24909535200

Correct answer:

24909535200

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 asked to approximate $\int_{3}^{15}x!dx$

So the interval is [3,15] , the subintervals have length $\frac{15-3}{3}=4$ , and since we are using the midpoints of each interval, the x-values are [5,9,13]

$\int_{3}^{15}x!dx \approx (4)[5!+9!+13!]$

$\int_{3}^{15}x!dx \approx 24909535200$

Report an Error

Example Question #131 : Differential Functions

Utilize the method of midpoint Riemann sums to approximate the average of $f(x)=\sqrt[4]{cos(x^2)}$ over the interval (-0.6,0.6) using four midpoints.

Possible Answers:

1.197

0.444

0.997

1.359

0.299

Correct answer:

0.997

Explanation:

To find the average of a function over a given interval of values [a,b] , the most precise method is to use an integral as follows:

$\frac{1}{b-a}\int_{a}^{b} f(x)dx$

Now for functions that are difficult or impossible to integrate,a Riemann sum can be used to approximate the value. 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 equal to the subinterval length $\frac{b-a}{n}$ , and variable heights f(x_i) , which depend on the function value at a given point x_i .

Now note that when using the method of Riemann sums to find an average value of a function, the expression changes:

$\frac{1}{b-a}\int_a^b f(x)dx \approx (\frac{1}{b-a})\frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))$

$\frac{1}{b-a}\int_a^b f(x)dx \approx \frac{1}{n}(f(x_1)+f(x_2)+...+f(x_n))$

We're asked to approximate the average of $f(x)=\sqrt[4]{cos(x^2)}$ over the interval (-0.6,0.6)

The subintervals have length $\frac{0.6-(-0.6)}{4}=0.3$ , and since we are using the midpoints of each interval, the x-values are [-0.45,-0.15,0.15,0.45]

${\frac{1}{1.2}\int_{-0.6}^{0.6}\sqrt[4]{cos(x^2)}dx\approx (\frac{1}{4})[\sqrt[4]{cos((-0.45)^2)}+\sqrt[4]{cos((-0.15)^2)}+\sqrt[4]{cos(0.15^2)}+\sqrt[4]{cos(0.45^2)}]}$

$\frac{1}{1.2}\int_{-0.6}^{0.6}\sqrt[4]{cos(x^2)}dx\approx 0.997$

Report an Error

View Calculus Tutors

Samantha
Certified Tutor

Middlesex County College, Associate in Science, Sustainability Studies.

View Calculus Tutors

Blossom
Certified Tutor

University of Maryland-Baltimore County, Bachelor of Science, Computer Science.

View Calculus Tutors

Abhisek
Certified Tutor

University of Florida, Bachelor of Engineering, Computer Science.

All Calculus 1 Resources

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

Popular Subjects

French Tutors in Seattle, ACT Tutors in San Diego, ISEE Tutors in Atlanta, Math Tutors in Dallas Fort Worth, Chemistry Tutors in Los Angeles, Spanish Tutors in Washington DC, Math Tutors in Washington DC, Spanish Tutors in Los Angeles, English Tutors in San Diego, Statistics Tutors in New York City

Popular Courses & Classes

GRE Courses & Classes in Dallas Fort Worth, GMAT Courses & Classes in San Diego, SSAT Courses & Classes in San Francisco-Bay Area, Spanish Courses & Classes in Phoenix, GRE Courses & Classes in Chicago, SAT Courses & Classes in New York City, GMAT Courses & Classes in New York City, GRE Courses & Classes in Boston, ACT Courses & Classes in San Diego, GMAT Courses & Classes in Dallas Fort Worth

Popular Test Prep

MCAT Test Prep in Boston, MCAT Test Prep in New York City, SSAT Test Prep in Denver, SAT Test Prep in Miami, GMAT Test Prep in Dallas Fort Worth, SSAT Test Prep in Washington DC, ACT Test Prep in San Diego, SAT Test Prep in Houston, ISEE Test Prep in Phoenix, GMAT Test Prep in Boston

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 #1151 : Calculus

Example Question #132 : Midpoint Riemann Sums

Example Question #133 : Midpoint Riemann Sums

Example Question #134 : Midpoint Riemann Sums

Example Question #135 : Midpoint Riemann Sums

Example Question #136 : Midpoint Riemann Sums

Example Question #137 : Midpoint Riemann Sums

Example Question #138 : Midpoint Riemann Sums

Example Question #139 : Midpoint Riemann Sums

Example Question #131 : Differential Functions

All Calculus 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: