Call Now to Set Up Tutoring:

(888) 888-0446

AP Calculus AB : Integrals

Study concepts, example questions & explanations for AP Calculus AB

Create An Account

All AP Calculus AB Resources

3 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

← Previous 1 2 3 4 5 6 7 8 9 10 … 31 32 Next →

Example Question #1 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

What is the domain of $f(x)=\frac{x+5}{\sqrt{x^2-9}}$ ?

Possible Answers:

$(-\infty,-3)\cup (3,+\infty)$

$(-\infty,-3]\cup [3,+\infty)$

$(-5,+\infty)$

$(3,+\infty)$

$(-\infty ,+\infty )$

Correct answer:

$(-\infty,-3)\cup (3,+\infty)$

Explanation:

${\sqrt{x^2-9}}>0$ because the denominator cannot be zero and square roots cannot be taken of negative numbers

$x^2-9>0$

$x^2>9$

$\sqrt{x^2}>\sqrt{9}$

$\left | x \right |>3$

$x>3\: or\, x<-3$

Report an Error

Example Question #6 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

If $y-6x-x^{2}=4$ ,

then at x=1 , what is 's instantaneous rate of change?

Possible Answers:

Correct answer:

Explanation:

The answer is 8.

$y-6x-x^{2}=4$

$y=x^{2}+6x+4$

$y'=2x+6$

$y^{'}=2(1)+6=8$

Report an Error

Example Question #1 : Interpretations And Properties Of Definite Integrals

Which of the following represents the graph of the polar function $r = 4\cos\theta$ in Cartestian coordinates?

Possible Answers:

(x-1)^2 +(y-2)^2 =4

(x-2)^2 +y^2 =4

x^2+(y-2)^2 =4

(x-2)^2 +(y-2)^2 =4

x^2 +y^2 =4

Correct answer:

(x-2)^2 +y^2 =4

Explanation:

$r = 4\cos\theta$

First, mulitply both sides by r.

$r^2 =4r\cos\theta$

Then, use the identities x^2+y^2 =r^2 and $x = r\cos\theta$ .

x^2+y^2=4x

x^2-4x+y^2=0

x^2 -4x + 4 +y^2 =4

(x-2)^2 +y^2 =4

The answer is (x-2)^2 +y^2 =4 .

Report an Error

Example Question #7 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

What is the average value of the function $f(x) = 3x^{2} - 5x +7$ from x = 1 to x=5 ?

Possible Answers:

Correct answer:

Explanation:

The average function value is given by the following formula:

$Ave = \frac{1}{5-1}\int_{1}^{5} f(x) dx = \frac{1}{5-1}\int_{1}^{5} (3x^{2} - 5x + 7)dx$

$Ave = \frac{1}{4} [x^{3}-\frac{5}{2}x^{2} + 7x]$ , evaluated from x = 1 to x = 5 .

$Ave = \frac{1}{4}[(125-\frac{125}{2} + 35) - (1-\frac{5}{2}+7)] = 23$

Report an Error

Example Question #8 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

$h(x)=\frac{g(x)}{(1+f(x))}$

If $f(x)=1,\ f^{'}(x)=2,\ g(x)=3,\ and\ g^{'}(x)=4$

then find $h^{'}(x)$ .

Possible Answers:

$\frac{1}{2}$

$\frac{5}{2}$

$\frac{-1}{2}$

Correct answer:

$\frac{1}{2}$

Explanation:

We see the answer is 0 after we do the quotient rule.

$h(x)=\frac{g(x)}{(1+f(x))}$

$h'(x)=\frac{g'(x)(1+f(x))-g(x)(f'(x))}{(1+f(x))^{2}}$

${h}'(x)=\frac{4(1+1)-3(2)}{(1+1)^{2}}=\frac{8-6}{4}=\frac{1}{2}$

Report an Error

Example Question #11 : Calculus 3

If $g(x)=\int_{0}^{x^2}f(t)dt$ , then which of the following is equal to $g'(x)$ ?

Possible Answers:

$f(x^{2})-f(0)$

$f(2x)$

$g(f(x^{2}))$

$2xf(x^{2})$

$f(x^{2})$

Correct answer:

$2xf(x^{2})$

Explanation:

According to the Fundamental Theorem of Calculus, if we take the derivative of the integral of a function, the result is the original function. This is because differentiation and integration are inverse operations.

For example, if $h(x)=\int_{a}^{x}f(u)du$ , where is a constant, then $h'(x)=f(x)$ .

We will apply the same principle to this problem. Because the integral is evaluated from 0 to $x^{2}$ , we must apply the chain rule.

$g'(x)=\frac{d}{dx}\int_{0}^{x^{2}}f(t)dt=f(x^{2})\cdot \frac{d}{dx}(x^{2})$

$=2xf(x^{2})$

The answer is $2xf(x^{2})$ .

Report an Error

Example Question #9 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

$\begin{align*}&\text{The velocity of a particle is given as:}\\&v(t)=\frac{(26sin(t + 2))}{3}\\&\text{Find its change in position over the time interval }t=4,5.5\end{align*}$

Possible Answers:

5.32

35.09

1.97

0.54

Correct answer:

5.32

Explanation:

$\begin{align*}&\text{If the velocity function of a particle is known, finding the}\\&\text{distance it travels over a given interval of time is only a}\\&\text{matter of integrating the velocity function over this same interval.}\\&\text{Considering the function given:}\\&\text{Utilizing integral rules:}\\&\int[sin(ax)]=-\frac{cos(ax)}{a}\\&\int_{4}^{5.5}(\frac{(26sin(t + 2))}{3})dt=-\frac{(26cos(t + 2))}{3}|_{4}^{5.5}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{4}^{5.5}(\frac{(26sin(t + 2))}{3})dt=-\frac{(26cos((5.5) + 2))}{3}-(-\frac{(26cos((4) + 2))}{3})\\&\int_{4}^{5.5}(\frac{(26sin(t + 2))}{3})dt=5.32\end{align*}$

Report an Error

Example Question #51 : Integrals

$\begin{align*}&\text{The velocity of a particle is given as:}\\&v(t)=\frac{21}{5}\\&\text{Find its change in position over the time interval }t=4,7\end{align*}$

Possible Answers:

3.50

1.85

1.45

12.60

Correct answer:

12.60

Explanation:

$\begin{align*}&\text{If the function defining a rate of change is known, the total}\\&\text{change over a given interval of time is only a matter of integrating}\\&\text{this rate function over this same time interval. Considering}\\&\text{the function given:}\\&\text{Utilizing integral rules:}\\&\int x^{a}=\frac{x^{a+1}}{a+1}\\&\int_{4}^{7}(\frac{21}{5})dt=\frac{(21t)}{5}|_{4}^{7}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{4}^{7}(\frac{21}{5})dt=\frac{(21(7))}{5}-(\frac{(21(4))}{5})\\&\int_{4}^{7}(\frac{21}{5})dt=12.60\end{align*}$

Report an Error

Example Question #11 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

$\begin{align*}&\text{The flow of water into a tank is given as:}\\&\frac{dV(t)}{dt}=\frac{(79e^{(-t)})}{9}\\&\text{Find the change in volume over the time interval }t=[0.5,1.5]\end{align*}$

Possible Answers:

3.37

10.10

31.97

0.50

Correct answer:

3.37

Explanation:

$\begin{align*}&\text{If the function defining a rate of change is known, the total}\\&\text{change over a given interval of time is only a matter of integrating}\\&\text{this rate function over this same time interval. Considering}\\&\text{the function given:}\\&\text{Utilizing integral rules:}\\&\int[e^{ax}]=\frac{e^{ax}}{a}\\&\int_{0.5}^{1.5}(\frac{(79e^{(-t)})}{9})dt=-\frac{(79e^{(-t)})}{9}|_{0.5}^{1.5}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{0.5}^{1.5}(\frac{(79e^{(-t)})}{9})dt=-\frac{(79e^{(-(1.5))})}{9}-(-\frac{(79e^{(-(0.5))})}{9})\\&\int_{0.5}^{1.5}(\frac{(79e^{(-t)})}{9})dt=3.37\end{align*}$

Report an Error

Example Question #11 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

$\begin{align*}&\text{The flow of water into a tank is given as:}\\&\frac{dV(t)}{dt}=\frac{(7\cdot 3^{(\frac{t}{3})})}{2}\\&\text{Find the change in volume over the time interval }t=[0.5,2]\end{align*}$

Possible Answers:

8.40

68.06

1.25

19.33

Correct answer:

8.40

Explanation:

$\begin{align*}&\text{If the function defining a rate of change is known, the total}\\&\text{change over a given interval of time is only a matter of integrating}\\&\text{this rate function over this same time interval. Considering}\\&\text{the function given:}\\&\text{Utilizing integral rules:}\\&\int[b^{ax}]=\frac{b^{ax}}{aln(b)}\\&\int_{0.5}^{2}(\frac{(7\cdot 3^{(\frac{t}{3})})}{2})dt=\frac{(21\cdot 3^{(\frac{t}{3})})}{(2ln(3))}|_{0.5}^{2}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{0.5}^{2}(\frac{(7\cdot 3^{(\frac{t}{3})})}{2})dt=\frac{(21\cdot 3^{(\frac{(2)}{3})})}{(2ln(3))}-(\frac{(21\cdot 3^{(\frac{(0.5)}{3})})}{(2ln(3))})\\&\int_{0.5}^{2}(\frac{(7\cdot 3^{(\frac{t}{3})})}{2})dt=8.40\end{align*}$

Report an Error

← Previous 1 2 3 4 5 6 7 8 9 10 … 31 32 Next →

View AP Calculus AB Tutors

Zhi Xuan
Certified Tutor

Queen's University, Bachelor, Engineering .

View AP Calculus AB Tutors

Sonu
Certified Tutor

Rajasthan Technical University Kota, Bachelor, Electrical.

View AP Calculus AB Tutors

Nina
Certified Tutor

Princeton University, Bachelor of Science, Computer Science.

All AP Calculus AB Resources

3 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Math Tutors in Philadelphia, Algebra Tutors in Chicago, MCAT Tutors in Washington DC, GRE Tutors in San Diego, English Tutors in Philadelphia, SAT Tutors in San Francisco-Bay Area, SSAT Tutors in Atlanta, ACT Tutors in San Diego, LSAT Tutors in San Francisco-Bay Area, MCAT Tutors in Atlanta

Popular Courses & Classes

ACT Courses & Classes in Philadelphia, Spanish Courses & Classes in Los Angeles, GRE Courses & Classes in Washington DC, GMAT Courses & Classes in Dallas Fort Worth, Spanish Courses & Classes in Boston, ISEE Courses & Classes in Houston, SAT Courses & Classes in New York City, GMAT Courses & Classes in Philadelphia, SAT Courses & Classes in San Francisco-Bay Area, ISEE Courses & Classes in Boston

Popular Test Prep

GRE Test Prep in Miami, GMAT Test Prep in San Diego, ISEE Test Prep in Houston, ISEE Test Prep in Seattle, SAT Test Prep in Washington DC, GMAT Test Prep in Denver, ACT Test Prep in Phoenix, ACT Test Prep in Philadelphia, LSAT Test Prep in Houston, LSAT 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

AP Calculus AB : Integrals

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #1 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

Example Question #6 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

Example Question #1 : Interpretations And Properties Of Definite Integrals

Example Question #7 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

Example Question #8 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

Example Question #11 : Calculus 3

Example Question #9 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

Example Question #51 : Integrals

Example Question #11 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

Example Question #11 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

All AP Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: