Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Introduction to Integrals

Study concepts, example questions & explanations for Calculus 2

Create An Account

All Calculus 2 Resources

9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #91 : Riemann Sums

Compute the following integral by using four equal left hand Riemann sums:

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

Possible Answers:

$\frac{64}{3}$

Correct answer:

Explanation:

Riemann sums are simply estimating the area under the curve of our function using an approximation of a rectangle. The area of a rectangle is given by A=length*width . Therefore, we need to figure both of these values for our four boxes and sum them all up. Since the boxes are equal, we can simply find the width by the following equation:

$width=\frac{b-a}{n}=\frac{4-0}{4}=1.$

and are your integration limits, and refers to the number of boxes.

The length of all of our boxes is given by the function. Since, this is a left Riemann sums, we need to consider the left side of each integration interval.

$\int_{0}^{4}x^2=0^2*1+1^2*1+2^2*1+3^2*1=0+1+4+9=14.$

For the lengths, we used -values of 0,1,2,3 , since those are the starting points of our four intervals.

Report an Error

Example Question #91 : Riemann Sums

Approximate the following integral with four equal right Riemann sums:

$\int_{2}^{10}x^3dx.\textup{}$

Possible Answers:

3584

1592

796

1792

Correct answer:

3584

Explanation:

$width=\frac{b-a}{n}=\frac{10-2}{4}=2.$

and are your integration limits, and refers to the number of boxes.

The length of all of our boxes is given by the function. Since, this is a right Riemann sums, we need to consider the right side of each integration interval.

$\int_{2}^{10}x^3=4^3*2+6^3*2+8^3*2+10^3*2$

=2(4^3+6^3+8^3+10^3)=2(1792)=3584.

For the lengths, we used -values of 4,6,8,10 , since those are the ending points of our four intervals.

Report an Error

Example Question #1891 : Calculus Ii

Find the result:

$\frac{\mathrm{d} }{\mathrm{d} x} \int_{1}^{e^x} t \ln t \; \mathrm{d}t$

Possible Answers:

$xe^{x }+ e$

$xe^{x+1}$

2xe^x

$xe^{2x}$

2xe^x + 1

Correct answer:

$xe^{2x}$

Explanation:

Set u = e^x . Then $\frac{\mathrm{d} u}{\mathrm{d} x} = e^x$ , and by the chain rule,

$\frac{\mathrm{d} }{\mathrm{d} x} \int_{1}^{e^x} t \ln t \; \mathrm{d}t$

$=\frac{\mathrm{d} u}{\mathrm{d} x} \cdot \frac{\mathrm{d} }{\mathrm{d} u} \int_{1}^{u} t \ln t \; \mathrm{d}t$

$= e^x \cdot \frac{\mathrm{d} }{\mathrm{d} u} \int_{1}^{u} t \ln t \; \mathrm{d}t$

By the fundamental theorem of Calculus, the above can be rewritten as

$e^x \cdot u \ln u$

$= e^x \cdot e^x \ln e^x$

$= e^{2x} \cdot x = xe^{2x}$

Report an Error

Example Question #1892 : Calculus Ii

Find the result:

$\frac{\mathrm{d} }{\mathrm{d} x} \int_{1}^{\ln x} t \log e^t \; \mathrm{d}t$

Possible Answers:

$\frac{\left (\ln x \right )^2}{x \ln 10}$

$\left (\ln x \right )^2$

$\frac{ 2 \ln x }{ \ln 10}$

$\frac{\left (\ln x \right )^2}{ \ln 10}$

$\frac{ 2 \ln x }{x \ln 10}$

Correct answer:

$\frac{\left (\ln x \right )^2}{x \ln 10}$

Explanation:

Set $u = \ln x$ . Then by the chain rule,

$\frac{\mathrm{d} }{\mathrm{d} x} \int_{1}^{\ln x} t \log e^t \; \mathrm{d}t$

$= \frac{\mathrm{d} u}{\mathrm{d} x} \cdot \frac{\mathrm{d} }{\mathrm{d} u} \int_{1}^{u} t \log e^t \; \mathrm{d}t$

$= \frac{1}{x} \cdot \frac{\mathrm{d} }{\mathrm{d} u} \int_{1}^{u} t \log e^t \; \mathrm{d}t$

By the Fundamental Theorem of Calculus, the above is equal to

$= \frac{1}{x} \cdot u \log e^u$

$= \frac{1}{x} \cdot \ln x \cdot \log e^ {\ln x}$

$= \frac{1}{x} \cdot \ln x \cdot \log x$

$= \frac{1}{x} \cdot \ln x \cdot \frac{\ln x}{\ln 10}$

$= \frac{\left (\ln x \right )^2}{x \ln 10}$

Report an Error

Example Question #1893 : Calculus Ii

Find the result:

$\frac{\mathrm{d} }{\mathrm{d} x}\int_{0}^{10x} \cos \frac{\pi t}{4} \log t \; \mathrm{d}t$

Possible Answers:

$10 \cos \frac{ 5\pi x}{2} \left (1 + \log x \right )$

$10 \cos \frac{ 5\pi x}{2}$

$10 \cos \frac{ 5\pi x}{2} \left (-1 + \log x \right )$

$10 \cos \frac{ 5\pi x}{2} \left (1 - \log x \right )$

$10 \cos \frac{ 5\pi x}{2} \log x$

Correct answer:

$10 \cos \frac{ 5\pi x}{2} \left (1 + \log x \right )$

Explanation:

Set u = 10x . Then $\frac{\mathrm{d} u}{\mathrm{d} x} = 10$ and by the chain rule,

$\frac{\mathrm{d} }{\mathrm{d} x}\int_{0}^{10x} \cos \frac{\pi t}{4} \log t \; \mathrm{d}t$

$=\frac{\mathrm{d} u }{\mathrm{d} x} \cdot \frac{\mathrm{d} }{\mathrm{d} u} \int_{0}^{u} \cos \frac{\pi t}{4}\log t \; \mathrm{d}t$

$=10 \cos \frac{\pi u}{4} \log u$

$=10 \cos \frac{\pi \cdot 10x}{4} \log 10x$

$=10 \cos \frac{ 5\pi x}{2} \log 10x$

$=10 \cos \frac{ 5\pi x}{2} \left (\log 10 + \log x \right )$

$=10 \cos \frac{ 5\pi x}{2} \left (1 + \log x \right )$

Report an Error

Example Question #1 : Fundamental Theorem Of Calculus And Techniques Of Antidifferentiation

Evaluate f'(0) :

$f(x)=\int_{0}^{x} \sin\left(\sqrt{t^2+\frac{\pi^2}{4}} \right )dt$

Possible Answers:

$-\frac{\pi}{2}$

$\frac{\pi}{2}$

Correct answer:

Explanation:

By the Fundamental Theorem of Calculus, we have that ${f'(x) = \frac{d}{dx} \int_0^x \sin \sqrt{t^2 + \frac{\pi ^2}{4}} dt = \sin \sqrt{x^2 + \frac{\pi ^2}{4}}$ . Thus, $f'(0) = \sin \sqrt{0^2 + \frac{\pi ^2}{4}} =1$ .

Report an Error

Example Question #1895 : Calculus Ii

Find the result:

$\frac{\mathrm{d} }{\mathrm{d} x} \int_{1}^{e^x} (\ln t)^3 \; \mathrm{d}t$

Possible Answers:

3x^2 e^x

$\frac{3e^x}{x^2 }$

$\frac{e^x}{3x^2 }$

x^3 e^x

$\frac{e^x}{x^3 }$

Correct answer:

x^3 e^x

Explanation:

Let u = e^x . Then $\frac{\mathrm{d} u}{\mathrm{d} x} = e^x$ , and by the chain rule,

$\frac{\mathrm{d} }{\mathrm{d} x} \int_{1}^{e^x} (\ln t)^3 \; \mathrm{d}t$

$=\frac{\mathrm{d}u }{\mathrm{d} x} \cdot \frac{\mathrm{d} }{\mathrm{d} u} \int_{1}^{u} (\ln t)^3 \; \mathrm{d}t$

$=\frac{\mathrm{d}u }{\mathrm{d} x} \cdot (\ln u)^3$

$=e^x \cdot (\ln e^x)^3$

$=e^x \cdot x^3 = x^3 e^x$

Report an Error

Example Question #1896 : Calculus Ii

Find the result:

$\frac{\mathrm{d} }{\mathrm{d} x}\int_{0}^{x^2} t^{3} \; \mathrm {d}t$

Possible Answers:

x^6

2x^8

2 x^6

x^7

2 x^7

Correct answer:

2 x^7

Explanation:

Set u = x^2 . Then $\frac{\mathrm{d}u }{\mathrm{d} x} = 2x$ .

By the chain rule,

$\frac{\mathrm{d} }{\mathrm{d} x}\int_{0}^{x^2} t^{3} \; \mathrm {d}t$

$= \frac{\mathrm{d}u }{\mathrm{d} x} \cdot \frac{\mathrm{d} } {\mathrm{d}u } \int_{0}^{u} t^{3} \; \mathrm {d}t$

$=2x \cdot \frac{\mathrm{d} } {\mathrm{d}u } \int_{0}^{u} t^{3} \; \mathrm {d}t$

By the Fundamental Theorem of Calculus, the above is equal to

$2x \cdot u^3 = 2x \cdot (x^2)^3 = 2x \cdot x^6 = 2 x^7$ .

Report an Error

Example Question #1897 : Calculus Ii

Find the result:

$\frac{\mathrm{d} }{\mathrm{d} x}\int_{0}^{4x} e^t \; \mathrm {d}t$

Possible Answers:

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

$\ln 4x$

$4 \ln x$

$e^{4x}$

$4e^{4x}$

Correct answer:

$4e^{4x}$

Explanation:

Set u = 4t . Then $\frac{\mathrm{d} u}{\mathrm{d} x} = 4$ .

By the chain rule,

$\frac{\mathrm{d} }{\mathrm{d} x}\int_{0}^{4x} e^t \; \mathrm {d}t$

$= \frac{\mathrm{d} u}{\mathrm{d} x} \cdot \frac{\mathrm{d} }{\mathrm{d} u}\int_{0}^{u} e^t \; \mathrm {d}t$

$= 4 \cdot \frac{\mathrm{d} }{\mathrm{d} u}\int_{0}^{u} e^t \; \mathrm {d}t$

By the Fundamental Theorem of Calculus, the above is equal to

$4 e^u = 4e^{4x}$ .

Report an Error

Example Question #1898 : Calculus Ii

Evaluate f'(4) when $f(x)=\int_{0}^{x}(2t^{2}+5t-4)dt$ .

Possible Answers:

Correct answer:

Explanation:

Via the Fundamental Theorem of Calculus, we know that, given a function $f(x)=\int_{0}^{x}(2t^{2}+5t-4)dt$ , $f'(x)=\frac{d}{dx}\int_{0}^{x}(2t^{2}+5t-4)dt=2x^{2}+5x-4$ .

Therefore $f'(4)=2(4)^{2}+5(4)-4=32+20-4=52-4=48$ .

Report an Error

View Calculus 2 Tutors

Bobby
Certified Tutor

University of Houston - Downtown, Bachelor of Science, Applied Mathematics.

View Calculus 2 Tutors

Victor
Certified Tutor

Rutgers University-New Brunswick, Bachelor of Science, Mathematics.

View Calculus 2 Tutors

Pankaj
Certified Tutor

University of Delhi, Electrical Engineer, Electrical Engineering. Columbia University in the City of New York, Master of Arts...

All Calculus 2 Resources

9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

GRE Tutors in Seattle, Spanish Tutors in Denver, Statistics Tutors in Philadelphia, ACT Tutors in San Diego, Algebra Tutors in Phoenix, Statistics Tutors in Houston, ACT Tutors in Seattle, GRE Tutors in Phoenix, SAT Tutors in Boston, ACT Tutors in Boston

Popular Courses & Classes

ACT Courses & Classes in New York City, GRE Courses & Classes in Philadelphia, ISEE Courses & Classes in Atlanta, MCAT Courses & Classes in Dallas Fort Worth, GMAT Courses & Classes in Miami, LSAT Courses & Classes in Philadelphia, GMAT Courses & Classes in New York City, ISEE Courses & Classes in Denver, SSAT Courses & Classes in San Diego, LSAT Courses & Classes in Atlanta

Popular Test Prep

ISEE Test Prep in Atlanta, SAT Test Prep in Boston, ACT Test Prep in San Diego, SAT Test Prep in San Diego, ISEE Test Prep in Washington DC, GRE Test Prep in Houston, GRE Test Prep in Miami, GMAT Test Prep in New York City, SAT Test Prep in Seattle, GMAT Test Prep in Denver

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 2 : Introduction to Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #91 : Riemann Sums

Example Question #91 : Riemann Sums

Example Question #1891 : Calculus Ii

Example Question #1892 : Calculus Ii

Example Question #1893 : Calculus Ii

Example Question #1 : Fundamental Theorem Of Calculus And Techniques Of Antidifferentiation

Example Question #1895 : Calculus Ii

Example Question #1896 : Calculus Ii

Example Question #1897 : Calculus Ii

Example Question #1898 : Calculus Ii

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: