Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Integral Applications

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 #41 : Area Under A Curve

Find the area bounded by the functions,

g(x)=x+6

f(x)=2(x+3)^2-1

Set up the integral and simplify the integrand.

Plot for area between curves problem

Possible Answers:

$A=\int_{-\frac{11}{4}-\frac{\sqrt{33}}{4}}^{\frac{11}{4}+\frac{\sqrt{33}}{4}}(2x^2+11x+11)dx$

$A=\int_{\frac{11}{4}-\frac{\sqrt{33}}{4}}^{\frac{11}{4}+\frac{\sqrt{33}}{4}}(2x^2+11x+11)dx$

$A=\int_{-\frac{11}{4}-\frac{\sqrt{33}}{4}}^{-\frac{11}{4}+\frac{\sqrt{33}}{4}}(-2x^2-11x-11)dx$

$A=\int_{\frac{7}{4}-\frac{\sqrt{209}}{4}}^{\frac{7}{4}+\frac{\sqrt{209}}{4}}(2x^2+11x+11)dx$

$A=\int_{-\frac{7}{2}-\frac{\sqrt{209}}{2}}^{-\frac{7}{2}+\frac{\sqrt{209}}{2}}(x^2+7x+7)dx$

Correct answer:

$A=\int_{-\frac{11}{4}-\frac{\sqrt{33}}{4}}^{-\frac{11}{4}+\frac{\sqrt{33}}{4}}(-2x^2-11x-11)dx$

Explanation:

Find the area bounded by the functions:

g(x)=x+6

f(x)=2(x+3)^2-1

Looking at the plot of the function we can see that $g(x)\geq f(x)$ for all in the region. The area formula is therefore,

$A=\int_{a}^{b}|g(x)-f(x)|dx$

, for g(a)=f(a) and g(b)=f(b)

To find the intersection points, set the functions equal to each other and solve for

g(x)=f(x)

x+6=2(x+3)^2-1

x+6=2(x^2+6x+9)-1

x+6=2x^2+12x+17

2x^2+11x+11=0

$x =\frac{-11\pm\sqrt{11^2-4(11)(2))}}{2(2)}$

$=-\frac{11}{4}\pm \frac{\sqrt{33}}{4}$

Therefore, the two lines intersect for the following values of :

$x = \left \{ -\frac{11}{4}-\frac{\sqrt{33}}{4},- \frac{11}{4}+\frac{\sqrt{33}}{4}\right \}$ .

So we will integrate over this interval,

$A=\int_{-\frac{11}{4}-\frac{\sqrt{33}}{4}}^{-\frac{11}{4}+\frac{\sqrt{33}}{4}}|(x+6)-[2(x+3)^2-1])dx$

$=\int_{-\frac{11}{4}-\frac{\sqrt{33}}{4}}^{-\frac{11}{4}+\frac{\sqrt{33}}{4}}[(x+6)-(2x^2+12x+17)]dx$

Finally we simplify the integrand to arrive at:

$=\int_{-\frac{11}{4}-\frac{\sqrt{33}}{4}}^{-\frac{11}{4}+\frac{\sqrt{33}}{4}}(-2x^2-11x-11)dx$

If you were to evaluate the integration is simple, but evaluating the limits would be quite tedious. The outcome is:

$=\left [ -\frac{2}{3}x^3-\frac{11}{2}x^2-11x\right ]_{\frac{11}{4}-\frac{\sqrt{33}}{4}}^{\frac{11}{4}+\frac{\sqrt{33}}{4}}$

$=\frac{11\sqrt{33}}{8}$

Report an Error

Example Question #41 : Area Under A Curve

What is the area under the curve f(x)=x^2-1 bounded by the x-axis from x=2 to x=4?

Possible Answers:

$\frac{50}{3}$

$\frac{51}{3}$

$\frac{49}{3}$

$-\frac{50}{3}$

Correct answer:

$\frac{50}{3}$

Explanation:

When setting up this problem, it should look like this: $\int_{2}^{4}x^2-1dx$ . Then, integrate. Remember that when integrating, raise the exponent by 1 and then also put that result on the denominator: $\frac{x^3}{3}-x$ . Then, evaluate first at 4 and then at 2. Subtract the results. $(\frac{64}{3}-4)-(\frac{8}{3}-2)=\frac{56}{3}-2=\frac{50}{3}$ .

Report an Error

Example Question #41 : Area Under A Curve

What is the area under the curve f(x)=x^2-3x-4 bounded by the x-axis from x=4 to x=5?

Possible Answers:

$\frac{19}{6}$

$\frac{17}{6}$

$\frac{21}{2}$

$\frac{17}{3}$

Correct answer:

$\frac{17}{6}$

Explanation:

First, set up the integral expression: $\int_{4}^{5}x^2-3x-4dx$ . Then, integrate. Remember, when integrating, raise the exponent by 1 and then put that result on the denominator: $\frac{x^3}{3}-\frac{3x^2}{2}-4x$ . Then, evaluate at 5 and then 4. Subtract those results: $(\frac{125}{3}-\frac{75}{2}-20)-(\frac{64}{3}-\frac{48}{2}-16)$ . Simplify to get your final answer of $\frac{17}{6}$ .

Report an Error

Example Question #102 : Integral Applications

What is the area under the curve f(x)=x^3-2x+1 bounded by the x-axis from x=2 to x=3?

Possible Answers:

$\frac{51}{4}$

$\frac{53}{4}$

$\frac{49}{2}$

$\frac{49}{4}$

Correct answer:

$\frac{49}{4}$

Explanation:

First, set up the integral expression: $\int_{2}^{3}x^3-2x+1dx$ . Then, integrate. Remember to raise the exponent by 1 and then put that result on the denominator: $\frac{x^4}{4}-\frac{2x^2}{2}+x$ . Evaluate at 3 and then 2. Subtract those 2 results: $(\frac{81}{4}-9+3)-(\frac{16}{4}-4+2)=\frac{65}{4}-4=\frac{49}{4}$ .

Report an Error

Example Question #271 : Integrals

What is the area under the curve f(x)=x^2-3x+1 bounded by the x-axis and from x=4 to x=5?

Possible Answers:

$\frac{47}{6}$

$\frac{7}{6}$

$\frac{49}{6}$

$\frac{52}{6}$

$\frac{53}{6}$

Correct answer:

$\frac{47}{6}$

Explanation:

First, set up the integral expression:
$\int_{4}^{5}x^2-3x+1dx$

Now, integrate:

$\frac{x^3}{3}-\frac{3x^2}{2}+x$

Evaluate at 5 and 4. Subtract the results:

$(\frac{125}{3}-\frac{75}{2}+5)-(\frac{64}{3}-\frac{48}{2}+4)=(\frac{125}{3}-\frac{75}{2}+5)$

Simplify to get:

$\frac{47}{6}$

Report an Error

Example Question #281 : Integrals

What is the area under the curve f(x)=2x^3-2x+1 bounded by the x-axis and from x=1 to x=2?

Possible Answers:

5.75

6.5

5.25

5.5

Correct answer:

5.5

Explanation:

First, set up the integral expression:

$\int_{1}^{2}2x^3-2x+1dx$

Now, integrate. Remember to raise the exponent by 1 and then also put that result on the denominator:

$\frac{2x^4}{4}-\frac{2x^2}{2}+x=\frac{x^4}{2}-x^2+x$

Evaluate at 2 and then 1. Subtract the results:

$(8-4+2)-(\frac{1}{2}-1+1)=6-\frac{1}{2}=5.5$

Report an Error

Example Question #44 : Area Under A Curve

What is the area under the curve f(x)=x^2-5x+4 bounded by the x-axis from x=4 to x=5?

Possible Answers:

$\frac{5}{3}$

$\frac{19}{6}$

$\frac{11}{6}$

$\frac{13}{6}$

$\frac{7}{6}$

Correct answer:

$\frac{11}{6}$

Explanation:

First, set up the integral expression:

$\int_{4}^{5}x^2-5x+4dx$

Now, integrate. Remember to raise the exponent by 1 and then also put that result on the denominator:

$\frac{x^3}{3}-\frac{5x^2}{2}+4x$

Evaluate at 5 and then 4. Subtract the results:

$(\frac{125}{3}-\frac{125}{2}+20)-(\frac{64}{3}-\frac{80}{2}+16)$

Simplify to get your answer of: $\frac{11}{6}$

Report an Error

Example Question #108 : Integral Applications

What is the area under the curve f(x)=x^2+4x+4 from x=4 to x=5 , bounded by the x-axis?

Possible Answers:

$\frac{8}{3}$

$\frac{1}{3}$

$\frac{11}{3}$

$\frac{2}{3}$

$\frac{5}{3}$

Correct answer:

$\frac{5}{3}$

Explanation:

First, set up the integral expression:

$\int_{4}^{5}x^2+4x+4dx$

Next, integrate. Remember to raise the exponent by 1 and also put that result on the denominator:

$\frac{x^3}{3}+\frac{4x^2}{2}+4x=\frac{x^3}{3}+2x^2+4x$

Now, evaluate at 5 and then 4. Subtract the results:

$(\frac{125}{3}+50+20)-(\frac{64}{3}+32+16)=22-\frac{61}{3}=\frac{5}{3}$

Report an Error

Example Question #281 : Integrals

What is the area under the curve f(x)=x^2-7x+9 bounded by the x-axis from x=0 to x=1?

Possible Answers:

$\frac{23}{6}$

$\frac{35}{6}$

$\frac{37}{6}$

$\frac{17}{6}$

$\frac{35}{2}$

Correct answer:

$\frac{35}{6}$

Explanation:

First, write out the integral expression:

$\int_{0}^{1}x^2-7x+9dx$

Next, integrate. Remember to add one to the exponent and also put that result on the denominator

$\frac{x^3}{3}-\frac{7x^2}{2}+9x$

Next, evaluate at 1 and then 0. Subtract the results:

$(\frac{1}{3}-\frac{7}{2}+9)-(0)=\frac{35}{6}$

Report an Error

Example Question #51 : Area Under A Curve

What is the area under the curve f(x)=x^3-2x^2+1 bounded by the x-axis from x=3 to x=4?

Possible Answers:

$\frac{233}{3}$

$\frac{233}{2}$

$\frac{233}{12}$

$\frac{233}{4}$

$\frac{133}{12}$

Correct answer:

$\frac{233}{12}$

Explanation:

First, write out the integral expression for this problem:

$\int_{3}^{4}x^3-2x^2+1dx$

Next, integrate. Remember to raise the exponent by 1 and also put that result on the denominator:

$\frac{x^4}{4}-\frac{2x^3}{3}+x$

Next, evaluate at 4 and then 3. Subtract the results:

$(\frac{256}{4}-\frac{128}{3}+4)-(\frac{81}{4}-\frac{54}{3}+3)$

Simplify to get your answer:

$\frac{175}{4}-\frac{74}{3}+1=\frac{233}{12}$

Report an Error

View Calculus 2 Tutors

Samita
Certified Tutor

University of Houston, Bachelor of Science, Computer Science. University of Houston, Non Degree Bachelors, Biology, General.

View Calculus 2 Tutors

Samantha
Certified Tutor

Middlesex County College, Associate in Science, Sustainability Studies.

View Calculus 2 Tutors

Satweka
Certified Tutor

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

All Calculus 2 Resources

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

Popular Subjects

SSAT Tutors in Philadelphia, SAT Tutors in Dallas Fort Worth, Statistics Tutors in Boston, GMAT Tutors in Phoenix, SSAT Tutors in Los Angeles, Physics Tutors in San Diego, Statistics Tutors in Philadelphia, MCAT Tutors in San Francisco-Bay Area, Reading Tutors in Philadelphia, French Tutors in Miami

Popular Courses & Classes

MCAT Courses & Classes in Los Angeles, SSAT Courses & Classes in New York City, ACT Courses & Classes in Seattle, SSAT Courses & Classes in Philadelphia, Spanish Courses & Classes in Chicago, ACT Courses & Classes in Dallas Fort Worth, ACT Courses & Classes in San Francisco-Bay Area, SAT Courses & Classes in Washington DC, GRE Courses & Classes in Atlanta, ISEE Courses & Classes in Seattle

Popular Test Prep

GRE Test Prep in Boston, GRE Test Prep in Atlanta, LSAT Test Prep in Washington DC, ACT Test Prep in San Francisco-Bay Area, ACT Test Prep in Phoenix, GRE Test Prep in Dallas Fort Worth, MCAT Test Prep in Seattle, SSAT Test Prep in Phoenix, GMAT Test Prep in Philadelphia, ACT 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 2 : Integral Applications

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #41 : Area Under A Curve

Example Question #41 : Area Under A Curve

Example Question #41 : Area Under A Curve

Example Question #102 : Integral Applications

Example Question #271 : Integrals

Example Question #281 : Integrals

Example Question #44 : Area Under A Curve

Example Question #108 : Integral Applications

Example Question #281 : Integrals

Example Question #51 : Area Under A Curve

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: