Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Definite 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 #231 : Definite Integrals

$\begin{align*}&\text{Evaluate the integral: }\\&\int_{7}^{11.5}(\frac{(9x^{2})}{40}- 2cos(x + 1))dx\end{align*}$

Possible Answers:

198.99

832.16

90.45

15.33

Correct answer:

90.45

Explanation:

$\begin{align*}&\text{In performing an integral, consider how in a derivative we’d}\\&\text{use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside. Integrals, in}\\&\text{similar fashion, entail a division. Now, although we have two}\\&\text{functions, since they're being subtracted (rather than multiplied/divided),}\\&\text{we integrate separately and combine the results. This is known}\\&\text{as superposition.}\\&\text{Utilizing integral rules:}\\&\int x^{a}=\frac{x^{a+1}}{a+1}\\&\int[cos(ax)]=\frac{sin(ax)}{a}\\&\int_{7}^{11.5}(\frac{(9x^{2})}{40}- 2cos(x + 1))dx=\frac{(3x^{3})}{40}- 2sin(x + 1)|_{7}^{11.5}\\&\text{Subtracting the function value at the lower bound from that of the upper bound:}\\&=\frac{(3(11.5)^{3})}{40}- 2sin((11.5) + 1)-(\frac{(3(7)^{3})}{40}- 2sin((7) + 1))\\&=90.45\end{align*}$

Report an Error

Example Question #232 : Definite Integrals

$\begin{align*}&\text{Evaluate the integral: }\\&\int_{-10}^{-8}(\frac{34}{(5x^{3})}-\frac{ (4e^{(-x)})}{55})dx\end{align*}$

Possible Answers:

-266.37

-157.40

-1385.15

-407.40

Correct answer:

-1385.15

Explanation:

$\begin{align*}&\text{In performing an integral, consider how in a derivative we’d}\\&\text{use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside. Integrals, in}\\&\text{similar fashion, entail a division. Now, although we have two}\\&\text{functions, since they're being subtracted (rather than multiplied/divided),}\\&\text{we integrate separately and combine the results. This is known}\\&\text{as superposition.}\\&\text{Utilizing integral rules:}\\&\int x^{a}=\frac{x^{a+1}}{a+1}\\&\int[e^{ax}]=\frac{e^{ax}}{a}\\&\int_{-10}^{-8}(\frac{34}{(5x^{3})}-\frac{ (4e^{(-x)})}{55})dx=\frac{(4e^{(-x)})}{55}-\frac{ 17}{(5x^{2})}|_{-10}^{-8}\\&\text{Subtracting the function value at the lower bound from that of the upper bound:}\\&=\frac{(4e^{(-(-8))})}{55}-\frac{ 17}{(5(-8)^{2})}-(\frac{(4e^{(-(-10))})}{55}-\frac{ 17}{(5(-10)^{2})})\\&=-1385.15\end{align*}$

Report an Error

Example Question #231 : Definite Integrals

$\begin{align*}&\text{Evaluate the integral: }\\&\int_{5}^{7.5}(\frac{(8\cdot 2^{(\frac{x}{3})})}{13}-\frac{ (8cos(3x))}{3})dx\end{align*}$

Possible Answers:

73.93

7.62

2.63

39.63

Correct answer:

7.62

Explanation:

$\begin{align*}&\text{In performing an integral, consider how in a derivative we’d}\\&\text{use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside. Integrals, in}\\&\text{similar fashion, entail a division. Now, although we have two}\\&\text{functions, since they're being subtracted (rather than multiplied/divided),}\\&\text{we integrate separately and combine the results. This is known}\end{align*}$ $\begin{align*}\\&\text{as superposition.}\\&\text{Utilizing integral rules:}\\&\int[b^{ax}]=\frac{b^{ax}}{aln(b)}\\&\int[cos(ax)]=\frac{sin(ax)}{a}\\&\int_{5}^{7.5}(\frac{(8\cdot 2^{(\frac{x}{3})})}{13}-\frac{ (8cos(3x))}{3})dx=\frac{(24\cdot 2^{(\frac{x}{3})})}{(13ln(2))}-\frac{ (8sin(3x))}{9}|_{5}^{7.5}\\&\text{Subtracting the function value at the lower bound from that of the upper bound:}\\&=\frac{(24\cdot 2^{(\frac{(7.5)}{3})})}{(13ln(2))}-\frac{ (8sin(3(7.5)))}{9}-(\frac{(24\cdot 2^{(\frac{(5)}{3})})}{(13ln(2))}-\frac{ (8sin(3(5)))}{9})\\&=7.62\end{align*}$

Report an Error

Example Question #232 : Definite Integrals

$\begin{align*}&\text{Evaluate the integral: }\\&\int_{8}^{9.5}(\frac{67}{x}-\frac{ 29}{(6\cdot 3^x)})dx\end{align*}$

Possible Answers:

5.23

1.19

11.51

57.57

Correct answer:

11.51

Explanation:

$\begin{align*}&\text{In performing an integral, consider how in a derivative we’d}\\&\text{use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside. Integrals, in}\\&\text{similar fashion, entail a division. Now, although we have two}\\&\text{functions, since they're being subtracted (rather than multiplied/divided),}\\&\text{we integrate separately and combine the results. This is known}\\&\text{as superposition.}\\&\text{Utilizing integral rules:}\\&\int x^{a}=\frac{x^{a+1}}{a+1}\\&\int[b^{ax}]=\frac{b^{ax}}{aln(b)}\\&\int_{8}^{9.5}(\frac{67}{x}-\frac{ 29}{(6\cdot 3^x)})dx=67ln(x) +\frac{ 29}{(6\cdot 3^xln(3))}|_{8}^{9.5}\\&\text{Subtracting the function value at the lower bound from that of the upper bound:}\\&=67ln((9.5)) +\frac{ 29}{(6\cdot 3^{(9.5)}ln(3))}-(67ln((8)) +\frac{ 29}{(6\cdot 3^{(8)}ln(3))})\\&=11.51\end{align*}$

Report an Error

Example Question #233 : Definite Integrals

$\int_{a}^{b}f(x)dx=3a+4b.$ What is $\int_{a}^{b}(f(x)+3)=?$

Possible Answers:

3a+4b+2

7b+6a

7ab

Correct answer:

Explanation:

For this problem, we need to first simplify the integral by splitting in two integrals. $\int_{a}^{b}(f(x)+3)dx=\int_{a}^{b}f(x)dx+\int_{a}^{b}3dx.$

Now, the first integral was given in the problem statement. Therefore, we just need to add that to the evaluation of the second integral.

$\int_{a}^{b}f(x)dx+\int_{a}^{b}3dx=3a+4b+\int_{a}^{b}3dx$

$\int_{a}^{b}3dx=3x=3(b)-3(a).$

Add the two parts together:

3a+4b+3b-3a=7b.

Report an Error

Example Question #234 : Definite Integrals

Evaluate this integral.

$\int_{0}^{\pi /2}cos^{3}(x)dx$

Possible Answers:

$\frac{1}{3}$

$-\frac{1}{3}$

$\frac{2}{3}$

$-\frac{2}{3}$

Correct answer:

$\frac{2}{3}$

Explanation:

$\int_{0}^{\pi /2}cos^{3}(x)dx=\int_{0}^{\pi /2}cos^{2}(x)*cos(x)dx =\int_{0}^{\pi /2}(1-sin^2(x))cos(x)dx$

Now let u=sin(x) thus du=cos(x)dx .

Also notice sin(0)=0 and $sin(\pi/2)=1$ .

So we have

$\int_{0}^{1}(1-u^2)du=u-\frac{1}{3}u^3$

evaluated between 0 and 1.

We have

$1-\frac{1}{3}*1^3=\frac{2}{3}$

Report an Error

Example Question #231 : Definite Integrals

Determine the area bounded by f(x)=x and g(x)=x^2 between on $0\leq x\leq 1$ .

Possible Answers:

$\frac{1}{3}$

$\frac{2}{3}$

$\frac{1}{6}$

$\frac{5}{6}$

$\frac{1}{2}$

Correct answer:

$\frac{1}{6}$

Explanation:

First identify the higher and lower function.

The higher function: f(x)=x

The lower function: g(x)=x^2

Since $f(x)\geq g(x)$ between x=0 and x=1 we must integrate.

To integrate take the difference of the lower function from the higher function.

$\int_{0}^{1}(x-x^2)dx=\frac{1}{2}x^2-\frac{1}{3}x^3$

Evaluating between zero and one results in,

$\\=\left[\frac{1}{2}1^2-\frac{1}{3}1^3\right]-\left[\frac{1}{2}0^2-\frac{1}{3}0^3 \right ] \\\\=\frac{1}{2}-\frac{1}{3}$

$=\frac{1}{6}$

Report an Error

Example Question #231 : Definite Integrals

Evaluate

$\int_{0}^{1} \frac{4}{1+x^2}dx$

Possible Answers:

$\frac{\pi }{2}$

$\pi$

$2\pi$

Correct answer:

$\pi$

Explanation:

$4*\int_{0}^{1}\frac{1}{1+x^2}dx = 4tan^{-1}(x)$

Evaluated between 0 and 1.

$=4*tan^{-1}(1)-4*tan^{-1}(0)=4*\frac{\pi }{4}-4*0=\pi$

Report an Error

Example Question #236 : Definite Integrals

$\int_{0}^{2}\sqrt{4-x^{^{2}}}dx$

Using geometry, Find the definite integral of the function.

Possible Answers:

$\frac{\pi}{2}$

$\pi$

$2\pi$

$4\pi$

Correct answer:

$\pi$

Explanation:

When we look at the figure we are trying to integrate, we see that it is actually a quarter circle.

Using the rules we learned in geometry, we know that the area of a circle is $\pi r^2$ . As such, we can take the radius and square it and then multiply it by $\pi$ . Our radius is found by taking the square root of the constant under the radical.

So now that we have the area of the whole circle, we need to divide it by 4.

$4\pi\div4=\pi$

Report an Error

Example Question #237 : Definite Integrals

$\int_{0}^{3}(3-x)dx$

Using geometry, solve the definite integral.

Possible Answers:

$\frac{9}{4}$

$\frac{9}{2}$

Correct answer:

$\frac{9}{2}$

Explanation:

When we look at the shape of the graph we notice that it is a triangle with a height of 3 and a base of 3.

So the area under the curve would be the same as the area of this triangle.

From geometry, from geometry we know that the area of a triangle is

$\frac{1}{2}\times (\text{Base})\times (\text{Height})$ .

So our answer would be

$\frac{1}{2}\times 3\times 3=\frac{9}{2}$

Report an Error

View Calculus 2 Tutors

Mohammed
Certified Tutor

University of Maryland-College Park, Bachelor of Science, Electrical Engineering. University of Maryland-College Park, Master...

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...

View Calculus 2 Tutors

Robert
Certified Tutor

University of Guelph, Bachelor, Biochemistry.

All Calculus 2 Resources

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

Popular Subjects

Statistics Tutors in Denver, GRE Tutors in Houston, ACT Tutors in Houston, LSAT Tutors in New York City, Calculus Tutors in New York City, LSAT Tutors in Washington DC, Biology Tutors in Houston, ISEE Tutors in Washington DC, Spanish Tutors in Boston, English Tutors in New York City

Popular Courses & Classes

ACT Courses & Classes in Philadelphia, Spanish Courses & Classes in Washington DC, Spanish Courses & Classes in Chicago, Spanish Courses & Classes in Miami, ACT Courses & Classes in Phoenix, ISEE Courses & Classes in Chicago, Spanish Courses & Classes in San Diego, ACT Courses & Classes in Chicago, Spanish Courses & Classes in Philadelphia, GRE Courses & Classes in Los Angeles

Popular Test Prep

ACT Test Prep in Philadelphia, GRE Test Prep in San Diego, ISEE Test Prep in San Diego, MCAT Test Prep in Miami, ACT Test Prep in Phoenix, LSAT Test Prep in Atlanta, SSAT Test Prep in Seattle, ACT Test Prep in Chicago, MCAT Test Prep in San Francisco-Bay Area, SSAT Test Prep in Miami

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 : Definite Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #231 : Definite Integrals

Example Question #232 : Definite Integrals

Example Question #231 : Definite Integrals

Example Question #232 : Definite Integrals

Example Question #233 : Definite Integrals

Example Question #234 : Definite Integrals

Example Question #231 : Definite Integrals

Example Question #231 : Definite Integrals

Example Question #236 : Definite Integrals

Example Question #237 : Definite Integrals

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: