We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

AP Calculus AB : AP Calculus AB

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

Example Question #33 : Fundamental Theorem Of Calculus

Evaluate the integral $\int_0^{\pi/2} \frac{\sin^2(x)}{\sec(x)}dx$

Possible Answers:

$\frac{1}{27}$

$\frac{1}{3}$

Correct answer:

$\frac{1}{3}$

Explanation:

To evaluate this integral, we must first rewrite the $\sec$ from the denominator, as a $\cos$ in the numerator. This gives us

$\int_0^{\pi/2} \sin^2(x)\cos(x)dx$

Since the $\sin$ has an exponent, we will pick it for our u-substitution.

$u = \sin(x)$

Now we find by differentiating.

$du = \cos(x)dx$

This gives us the cosine piece. Writing everything in terms or u, we get

$\int_{u_a}^{u_b} u^2du$

To integrate this we use the following basic integral form: $\int_a^b u^n du = [\frac{1}{n+1}u^{n+1}]|_a^b$

Applying this to our integral, we get

$[\frac{1}{3}u^3]$

Now we can "un-substitute" to get back to x-terms. Replacing with $\sin(x)$ , we get

$[\frac{1}{3}\sin^3(x)]|_0^{\pi/2}$

Now we use the 2nd Fundamental Theorem of Calculus. We plug in the upper bound of $\pi/2$ , then plug in the lower bound of , and subtract.

Doing this we get

$[\frac{1}{3}\sin^3(\pi/2)]-[\frac{1}{3}\sin^3(0)]$

$[\frac{1}{3}(1^3)]-[\frac{1}{3}(0^3)]$

$\frac{1}{3}(1)-\frac{1}{3}(0)$

$\frac{1}{3}$

This is the correct answer.

Report an Error

Example Question #1 : Basic Properties Of Definite Integrals (Additivity And Linearity)

If f(x),g(x) are continuous functions, $g(x) = \frac{1}{2}f(x)$ , $\int_0^4 f(x) dx = 6$ , and $\int_4^{10} g(x)dx = 8$ , find $\int_0^{10} f(x) dx$ .

Possible Answers:

Not enough information

Correct answer:

Explanation:

We proceed as follows

$\int_0^{10} f(x) dx$ . (Start)

$=\int_0^4 f(x) dx + \int_4^{10}f(x)dx$ . (Break up the integral using the additive rule.)

$=\int_0^4 f(x) dx + \int_4^{10}2g(x)dx$ . (We don't have information about the 2nd integral, so we solve our first equation for f(x) and replace it in this integral.)

$=\int_0^4 f(x) dx + 2\int_4^{10}g(x)dx$ . (Factor out the using linearity.)

=6 +2(8) . (Substitute in what we were given.)

=22 .

Report an Error

Example Question #1 : Basic Properties Of Definite Integrals (Additivity And Linearity)

$\begin{align*}&\int_{24}^{9}f(x)dx=-24\\&\int_{24}^{26}f(x)dx=11\\&\int_{26}^{32}f(x)dx=24\\&\int_{32}^{34}f(x)dx=34\\&\text{Calculate from the above values }\int_{9}^{34}f(x)dx\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{One of the principles of integrals is one of summation. If the}\\&\text{values of an integral across certain sets of points is known,}\\&\text{this could potentially be used to find integral values at other}\\&\text{points. Imagine a set of points that are in ascending numerical}\\&\text{value: a, b, c, and d. It follows that:}\\&\int_a^d=\int_a^b+\int_b^c+\int_c^d\\&\text{Another principle of integrals is that if an integral value}\\&\text{is known, to take the same integral backwards will give a negative}\\&\text{of the original value:}\\&\int_a^b=-\int_b^a\\&\text{Knowing this, we can calculate }\int_{9}^{34}f(x)dx\\&\int_{9}^{34}f(x)dx=-(-24)+(11)+(24)+(34)\\&\int_{9}^{34}f(x)dx=93\\&\end{align*}$

Report an Error

Example Question #1 : Basic Properties Of Definite Integrals (Additivity And Linearity)

$\begin{align*}&\int_{13}^{2}f(x)dx=-94\\&\int_{13}^{19}f(x)dx=-23\\&\int_{23}^{19}f(x)dx=-32\\&\int_{28}^{23}f(x)dx=-18\\&\int_{31}^{28}f(x)dx=-84\\&\text{Determine }\int_{2}^{31}f(x)dx\end{align*}$

Possible Answers:

205

178

Correct answer:

205

Explanation:

$\begin{align*}&\text{One of the principles of integrals is one of summation. If the}\\&\text{values of an integral across certain sets of points is known,}\\&\text{this could potentially be used to find integral values at other}\\&\text{points. Imagine a set of points that are in ascending numerical}\\&\text{value: a, b, c, and d. It follows that:}\\&\int_a^d=\int_a^b+\int_b^c+\int_c^d\\&\text{Another principle of integrals is that if an integral value}\\&\text{is known, to take the same integral backwards will give a negative}\\&\text{of the original value:}\\&\int_a^b=-\int_b^a\\&\text{Knowing this, we can calculate }\int_{2}^{31}f(x)dx\\&\int_{2}^{31}f(x)dx=-\int_{13}^{2}f(x)dx+\int_{13}^{19}f(x)dx-\int_{23}^{19}f(x)dx-\int_{28}^{23}f(x)dx-\int_{31}^{28}f(x)dx\\&\int_{2}^{31}f(x)dx=-(-94)+(-23)-(-32)-(-18)-(-84)\\&\int_{2}^{31}f(x)dx=205\\&\end{align*}$

Report an Error

Example Question #1 : Basic Properties Of Definite Integrals (Additivity And Linearity)

$\begin{align*}&\int_{-10}^{-5}f(x)dx=95\\&\int_{-1}^{-5}f(x)dx=-41\\&\int_{6}^{-1}f(x)dx=-26\\&\int_{21}^{6}f(x)dx=-38\\&\int_{28}^{21}f(x)dx=62\\&\text{Calculate from the above values }\int_{-10}^{28}f(x)dx\end{align*}$

Possible Answers:

138

Correct answer:

138

Explanation:

$\begin{align*}&\text{One of the principles of integrals is one of summation. If the}\\&\text{values of an integral across certain sets of points is known,}\\&\text{this could potentially be used to find integral values at other}\\&\text{points. Imagine a set of points that are in ascending numerical}\\&\text{value: a, b, c, and d. It follows that:}\\&\int_a^d=\int_a^b+\int_b^c+\int_c^d\\&\text{Another principle of integrals is that if an integral value}\\&\text{is known, to take the same integral backwards will give a negative}\\&\text{of the original value:}\\&\int_a^b=-\int_b^a\\&\text{Knowing this, we can calculate }\int_{-10}^{28}f(x)dx\\&\int_{-10}^{28}f(x)dx=\int_{-10}^{-5}f(x)dx-\int_{-1}^{-5}f(x)dx-\int_{6}^{-1}f(x)dx-\int_{21}^{6}f(x)dx-\int_{28}^{21}f(x)dx\\&\int_{-10}^{28}f(x)dx=(95)-(-41)-(-26)-(-38)-(62)\\&\int_{-10}^{28}f(x)dx=138\\&\end{align*}$

Report an Error

Example Question #31 : Interpretations And Properties Of Definite Integrals

$\begin{align*}&\int_{-3}^{9}f(x)dx=-67\\&\int_{9}^{22}f(x)dx=74\\&\int_{37}^{22}f(x)dx=-18\\&\int_{39}^{37}f(x)dx=-67\\&\int_{42}^{39}f(x)dx=68\\&\int_{42}^{45}f(x)dx=86\\&\text{Calculate from the above values }\int_{-3}^{45}f(x)dx\end{align*}$

Possible Answers:

110

Correct answer:

110

Explanation:

$\begin{align*}&\text{One of the principles of integrals is one of summation. If the}\\&\text{values of an integral across certain sets of points is known,}\\&\text{this could potentially be used to find integral values at other}\\&\text{points. Imagine a set of points that are in ascending numerical}\\&\text{value: a, b, c, and d. It follows that:}\\&\int_a^d=\int_a^b+\int_b^c+\int_c^d\\&\text{Another principle of integrals is that if an integral value}\\&\text{is known, to take the same integral backwards will give a negative}\\&\text{of the original value:}\\&\int_a^b=-\int_b^a\\&\text{Knowing this, we can calculate }\int_{-3}^{45}f(x)dx\\&\int_{-3}^{45}f(x)dx=\int_{-3}^{9}f(x)dx+\int_{9}^{22}f(x)dx-\int_{37}^{22}f(x)dx-\int_{39}^{37}f(x)dx-\int_{42}^{39}f(x)dx+\int_{42}^{45}f(x)dx\\&\int_{-3}^{45}f(x)dx=(-67)+(74)-(-18)-(-67)-(68)+(86)\\&\int_{-3}^{45}f(x)dx=110\\&\end{align*}$

Report an Error

Example Question #1 : Basic Properties Of Definite Integrals (Additivity And Linearity)

$\begin{align*}&\int_{-9}^{0}f(x)dx=62\\&\int_{13}^{0}f(x)dx=-46\\&\int_{18}^{13}f(x)dx=20\\&\text{Use the above values to find }\int_{-9}^{18}f(x)dx\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{One of the principles of integrals is one of summation. If the}\\&\text{values of an integral across certain sets of points is known,}\\&\text{this could potentially be used to find integral values at other}\\&\text{points. Imagine a set of points that are in ascending numerical}\\&\text{value: a, b, c, and d. It follows that:}\\&\int_a^d=\int_a^b+\int_b^c+\int_c^d\\&\text{Another principle of integrals is that if an integral value}\\&\text{is known, to take the same integral backwards will give a negative}\\&\text{of the original value:}\\&\int_a^b=-\int_b^a\\&\text{Knowing this, we can calculate }\int_{-9}^{18}f(x)dx\\&\int_{-9}^{18}f(x)dx=\int_{-9}^{0}f(x)dx-\int_{13}^{0}f(x)dx-\int_{18}^{13}f(x)dx\\&\int_{-9}^{18}f(x)dx=(62)-(-46)-(20)\\&\int_{-9}^{18}f(x)dx=88\\&\end{align*}$

Report an Error

Example Question #1 : Basic Properties Of Definite Integrals (Additivity And Linearity)

$\begin{align*}&\int_{-10}^{3}f(x)dx=95\\&\int_{3}^{13}f(x)dx=63\\&\int_{23}^{13}f(x)dx=78\\&\int_{23}^{37}f(x)dx=-54\\&\int_{37}^{50}f(x)dx=-56\\&\int_{50}^{56}f(x)dx=77\\&\text{Use the above values to find }\int_{-10}^{56}f(x)dx\end{align*}$

Possible Answers:

203

172

Correct answer:

Explanation:

$\begin{align*}&\text{One of the principles of integrals is one of summation. If the}\\&\text{values of an integral across certain sets of points is known,}\\&\text{this could potentially be used to find integral values at other}\\&\text{points. Imagine a set of points that are in ascending numerical}\\&\text{value: a, b, c, and d. It follows that:}\\&\int_a^d=\int_a^b+\int_b^c+\int_c^d\\&\text{Another principle of integrals is that if an integral value}\\&\text{is known, to take the same integral backwards will give a negative}\\&\text{of the original value:}\\&\int_a^b=-\int_b^a\\&\text{Knowing this, we can calculate }\int_{-10}^{56}f(x)dx\\&\int_{-10}^{56}f(x)dx=\int_{-10}^{3}f(x)dx+\int_{3}^{13}f(x)dx-\int_{23}^{13}f(x)dx+\int_{23}^{37}f(x)dx+\int_{37}^{50}f(x)dx+\int_{50}^{56}f(x)dx\\&\int_{-10}^{56}f(x)dx=(95)+(63)-(78)+(-54)+(-56)+(77)\\&\int_{-10}^{56}f(x)dx=47\\&\end{align*}$

Report an Error

Example Question #5 : Basic Properties Of Definite Integrals (Additivity And Linearity)

$\begin{align*}&\int_{-4}^{6}f(x)dx=5\\&\int_{12}^{6}f(x)dx=90\\&\int_{15}^{12}f(x)dx=73\\&\int_{19}^{15}f(x)dx=-70\\&\int_{22}^{19}f(x)dx=97\\&\int_{24}^{22}f(x)dx=36\\&\text{Calculate from the above values }\int_{-4}^{24}f(x)dx\end{align*}$

Possible Answers:

231

Correct answer:

Explanation:

$\begin{align*}&\text{One of the principles of integrals is one of summation. If the}\\&\text{values of an integral across certain sets of points is known,}\\&\text{this could potentially be used to find integral values at other}\\&\text{points. Imagine a set of points that are in ascending numerical}\\&\text{value: a, b, c, and d. It follows that:}\\&\int_a^d=\int_a^b+\int_b^c+\int_c^d\\&\text{Another principle of integrals is that if an integral value}\\&\text{is known, to take the same integral backwards will give a negative}\\&\text{of the original value:}\\&\int_a^b=-\int_b^a\\&\text{Knowing this, we can calculate }\int_{-4}^{24}f(x)dx\\&\int_{-4}^{24}f(x)dx=\int_{-4}^{6}f(x)dx-\int_{12}^{6}f(x)dx-\int_{15}^{12}f(x)dx-\int_{19}^{15}f(x)dx-\int_{22}^{19}f(x)dx-\int_{24}^{22}f(x)dx\\&\int_{-4}^{24}f(x)dx=(5)-(90)-(73)-(-70)-(97)-(36)\\&\int_{-4}^{24}f(x)dx=-221\\&\end{align*}$

Report an Error

Example Question #33 : Interpretations And Properties Of Definite Integrals

$\begin{align*}&\int_{-4}^{0}f(x)dx=-13\\&\int_{15}^{0}f(x)dx=56\\&\int_{15}^{23}f(x)dx=-53\\&\int_{23}^{31}f(x)dx=96\\&\int_{31}^{32}f(x)dx=-44\\&\text{Calculate from the above values }\int_{-4}^{32}f(x)dx\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{One of the principles of integrals is one of summation. If the}\\&\text{values of an integral across certain sets of points is known,}\\&\text{this could potentially be used to find integral values at other}\\&\text{points. Imagine a set of points that are in ascending numerical}\\&\text{value: a, b, c, and d. It follows that:}\\&\int_a^d=\int_a^b+\int_b^c+\int_c^d\\&\text{Another principle of integrals is that if an integral value}\\&\text{is known, to take the same integral backwards will give a negative}\\&\text{of the original value:}\\&\int_a^b=-\int_b^a\\&\text{Knowing this, we can calculate }\int_{-4}^{32}f(x)dx\\&\int_{-4}^{32}f(x)dx=\int_{-4}^{0}f(x)dx-\int_{15}^{0}f(x)dx+\int_{15}^{23}f(x)dx+\int_{23}^{31}f(x)dx+\int_{31}^{32}f(x)dx\\&\int_{-4}^{32}f(x)dx=(-13)-(56)+(-53)+(96)+(-44)\\&\int_{-4}^{32}f(x)dx=-70\\&\end{align*}$

Report an Error

View AP Calculus AB Tutors

Steven
Certified Tutor

University of Louisville, Bachelor of Science, Mechanical Engineering. University of Phoenix-Louisville Campus, Master of Eng...

View AP Calculus AB Tutors

MrG
Certified Tutor

Long Island University-Brooklyn Campus, Master of Science, Applied Mathematics.

View AP Calculus AB Tutors

Magdy
Certified Tutor

Cairo University, Egypt, Bachelor of Science, Electrical Engineering. New Mexico State University-Main Campus, Doctor of Phil...

All AP Calculus AB Resources

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

Popular Subjects

GMAT Tutors in Chicago, SSAT Tutors in Seattle, MCAT Tutors in San Francisco-Bay Area, Math Tutors in San Francisco-Bay Area, English Tutors in New York City, MCAT Tutors in Atlanta, Spanish Tutors in Seattle, ACT Tutors in Atlanta, GRE Tutors in Seattle, Calculus Tutors in Boston

Popular Courses & Classes

SSAT Courses & Classes in Houston, GRE Courses & Classes in Philadelphia, ACT Courses & Classes in San Francisco-Bay Area, Spanish Courses & Classes in Houston, ACT Courses & Classes in Philadelphia, SSAT Courses & Classes in Philadelphia, ACT Courses & Classes in Seattle, SSAT Courses & Classes in Los Angeles, ACT Courses & Classes in Boston, Spanish Courses & Classes in Phoenix

Popular Test Prep

SSAT Test Prep in Phoenix, ACT Test Prep in Washington DC, ACT Test Prep in Boston, SSAT Test Prep in Boston, ISEE Test Prep in Los Angeles, MCAT Test Prep in Chicago, ACT Test Prep in Atlanta, ACT Test Prep in Seattle, ACT Test Prep in Philadelphia, SAT Test Prep in San Diego

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 : AP Calculus AB

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #33 : Fundamental Theorem Of Calculus

Example Question #1 : Basic Properties Of Definite Integrals (Additivity And Linearity)

Example Question #1 : Basic Properties Of Definite Integrals (Additivity And Linearity)

Example Question #1 : Basic Properties Of Definite Integrals (Additivity And Linearity)

Example Question #1 : Basic Properties Of Definite Integrals (Additivity And Linearity)

Example Question #31 : Interpretations And Properties Of Definite Integrals

Example Question #1 : Basic Properties Of Definite Integrals (Additivity And Linearity)

Example Question #1 : Basic Properties Of Definite Integrals (Additivity And Linearity)

Example Question #5 : Basic Properties Of Definite Integrals (Additivity And Linearity)

Example Question #33 : Interpretations And Properties Of Definite Integrals

All AP Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: