We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

Create An Account

All Calculus 1 Resources

10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #29 : How To Find Area Of A Region

A curve is defined by the function:

$y=3x^3 - 4x^2 +8x +e^{2x}$

Find the area between the curve and the -axis on the interval of x = 0 to .

Possible Answers:

2e^2-4

$\frac{35}{12}$

$\frac{35+6e^2}{12}$

Correct answer:

$\frac{35+6e^2}{12}$

Explanation:

To find the area between the curve and the x-axis, we will need to integrate the curve with respect to x:

$y=3x^3 - 4x^2 +8x +e^{2x}$

$Y = \frac{3x^4}{4}-\frac{4x^3}{3}+4x^2+\frac{e^{2x}}{2}$ from x = 0 to 1

Introducing are upper and lower bounds, we can find the area to be:

$Area =\left(\frac{3}{4}-\frac{4}{3}+4+\frac{e^2}{2}\right) - \left(0-0+0+\frac{1}{2}\right)=\frac{35+6e^2}{12}$

Report an Error

Example Question #2991 : Functions

Find the area under the curve $5 x^{2} + e^{x}$ from x=0 to x= 4 , rounded to the nearest integer.

Possible Answers:

157

356

106

160

Correct answer:

160

Explanation:

Finding the area under the curve can be understood as taking the integral of the equation and it can be rewritten into the following:

$\int_{0}^{4} 5x^{2}+e^{x} dx$

To solve:

1. Find the indefinite integral of the function.

2. Plug in the upper and lower limit values and take the difference of the two.

1. Using the power rule which states,

$\int x^n=\frac{x^{n+1}}{n+1}$ to each term we find,

$\int 5x^{2} + e^{x} dx = \frac{5}{3}x^{3} + e^{x} + C$ .

2. $\left( \frac{5}{3}4^{3} + e^{4} \right) - \left( \frac{5}{3}0^{3} + e^{0} \right) = \frac{320}{3} + e^4 - 1 = 160.26$ , which is 160 after rounding.

Report an Error

Example Question #4021 : Calculus

Find the area under the curve $x^{5} + 2x^{2} + 4$ from x = 0 to x = 3 , rounded to the nearest integer.

Possible Answers:

143

152

164

176

157

Correct answer:

152

Explanation:

Finding the area under the curve can be understood as taking the integral of the equation and it can be rewritten into the following:

$\int_{0}^{3} x^{5} + 2x^{2} + 4dx$

To solve:

1. Find the indefinite integral of the function.

2. Plug in the upper and lower limit values and take the difference of the 2 values.

1. Using the power rule which states,

$\int x^n=\frac{x^{n+1}}{n+1}$ to each term we find,

$\int x^5 + 2x^2 + 4dx = \frac{x^6}{6} + \left(\frac{2}{3}\right)x^{3} + 4x + constant$

2. Plug in 3 and 0 for x and then take the difference.

$3^{6} + \left(\frac{2}{3}\right)\cdot 3^{3} + 4\cdot3 - 0$ = 151.50 , which rounds up to 152 .

Report an Error

Example Question #32 : How To Find Area Of A Region

Find the area under the curve $(x - 3)^{2} + 2x$ from x = 1 to x= 3 , rounded to the nearest integer.

Possible Answers:

Correct answer:

Explanation:

Finding the area under the curve can be understood as taking the integral of the equation and it can be rewritten into the following:

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

To solve:

1. Find the indefinite integral of the function.

2. Plug in the upper and lower limit values and take the difference of the two values.

1. Using the power rule which states,

$\int x^n=\frac{x^{n+1}}{n+1}$ to each term we find,

$\int (x-3)^2 + 2xdx = \frac{x^{3}}{3} - 2x^{2} + 9x$ .

2. Plug in 4 and 1 for x and then take the difference.

$\left(\frac{x^3}{3} - 2\cdot3^{2} + 9\cdot3\right) - \left(\frac{1}{3} - 2\cdot1^{2} + 9\right) = 10.667$

The answer is 11 because 10.667 rounds up.

Report an Error

Example Question #31 : Area

Find the area of the curve sin(x) + 2x from x = 0 to x = $\pi$

Possible Answers:

$\pi$

$1 + \pi^{2}$

$2 + \pi$

$2 + \pi ^{2}$

Correct answer:

$2 + \pi ^{2}$

Explanation:

Written in words, solve:

$\int_{0}^{\pi} sin(x) + 2x dx$

To solve:

1. Find the indefinite integral of the function.

2. Plug in the upper and lower limit values and take the difference of the two values.

1. Using the power rule which states,

$\int x^n=\frac{x^{n+1}}{n+1}$ to the term and recalling the integral of sin(x) is -cos(x) we find,

$\int sin(x) + 2xdx = x^{2} - cos(x)$ .

2. Plug in $\pi$ and for and then take the difference.

$\pi^{2} - cos(\pi) -(0- cos(0) )$ = $\pi^{2} + 2$

note:

$cos(0) = 1 , cos(\pi) = -1$

Report an Error

Example Question #32 : Area

Find the area under the curve $cos(x) + e^{x}$ from x = 0 to x = $\pi$ .

Possible Answers:

$e^{\pi-1}$

$e^{\pi}$

$e^{\pi} + 1$

$e^{\pi}-1$

Correct answer:

$e^{\pi}-1$

Explanation:

Finding the area under the curve can be understood as taking the integral of the equation and it can be rewritten into the following:

$\int_{0}^{\pi} cos(x) + e^{x}dx$

To solve:

1. Find the indefinite integral of the function.

2. Plug in the upper and lower limit values and take the difference of the two values.

1. Recall that the integral of cos(x) is sin(x) and that the integral of e^x is itself, we find that,

$\int cos(x) + e^{x}dx = e^{x} + sin(x)$ .

2. $e^{\pi} + sin(\pi) - (e^{0} + sin(0))$ $= e^{\pi}-1$

note: $sin(\pi) = sin(\pi) = 0$

Report an Error

Example Question #34 : How To Find Area Of A Region

Find the area under the curve cos(x) + 6 from x = 0 to x = $\pi$ .

Possible Answers:

$6\pi$

$\pi-6$

$6\pi^{2}$

$\pi + 6$

Correct answer:

$6\pi$

Explanation:

Finding the area under the curve can be understood as taking the integral of the equation and it can be rewritten into the following:

$\int_{0}^{\pi}cos(x) + 6 dx$

To solve:

1. Find the indefinite integral of the function.

2. Plug in the upper and lower limit values and take the difference of the two values.

1. Using the power rule which states,

$\int x^n=\frac{x^{n+1}}{n+1}$ to the term and recalling the integral of cos(x) is sin(x) we find,

$\int cos(x) + 6dx = sin(x)+6x+constant$

2. $sin(\pi)+6\pi - (sin(0) + 6\cdot0) = 6\pi$

Report an Error

Example Question #111 : Regions

Find the area under the curve cos(x)+2x from x=0 to $2\pi$ .

Possible Answers:

$\pi$

$4\pi^{2}$

$2\pi$

$2\pi^{2}$

Correct answer:

$4\pi^{2}$

Explanation:

Finding the area under the curve can be understood as taking the integral of the equation and it can be rewritten into the following:

$\int_0^{2\pi}cos(x)+2xdx$

To solve:

1. Find the indefinite integral of the function

2. Plug in the upper and lower limit values and take the difference of the 2 values.

1. Using the power rule which states,

$\int x^n=\frac{x^{n+1}}{n+1}$ to the term and recalling the integral of cos(x) is sin(x) we find,

$\int cos(x)+2xdx = x^{2}+sin(x)+constant$ .

2. $(2\pi)^{2}+sin(2\pi) - (0^2 + sin(0)) = (2\pi)^2 = 4\pi^2$

Report an Error

Example Question #35 : How To Find Area Of A Region

Find the area under the curve e^2 - 2x from x=0 to x=2 .

Possible Answers:

e^2 + 1

2e^2 - 4

e^2

e^2 - 4

e^2 -1

Correct answer:

2e^2 - 4

Explanation:

Finding the area under the curve can be understood as taking the integral of the equation and it can be rewritten into the following:

$\int_{0}^{2} e^2 - 2xdx$

To solve:

1. Find the indefinite integral of the function.

2. Plug in the upper and lower limit values and take the difference of the two values.

1. Using the power rule which states,

$\int x^n=\frac{x^{n+1}}{n+1}$ to each term we find,

$\int e^2 - 2xdx = xe^2 - x^2 + constant$ .

2. (2)e^2 - 2^2 - (0e^2 - 0^2) = 2e^2 - 4

Report an Error

Example Question #36 : How To Find Area Of A Region

Find the area of the region bounded by y = x^3 and y =x on the interval 0 < x < 1 .

Possible Answers:

0.25

1.25

0.75

0.5

Correct answer:

0.25

Explanation:

Finding the area of a region can be understood as taking the integral of the difference of the two equations and it can be rewritten into the following:

$\int_{0}^{1} x - x^3dx$

To determine which equation gets subtracted from, try drawing the graph or testing which function is larger at a point in the interval. The function that is larger will be on top, which in this case is y=x . For example, at point x =0.2 , y=x is 0.2 while y=x^3 is 0.008 .

$\int_{0}^{1} x - x^3dx = \frac{x^2}{2} - \frac{x^4}{4} |_0^{1} = \left(\frac{1}{2} - \frac{1}{4}\right) - (0-0) = \frac{1}{4} = 0.25$

Report an Error

View Calculus Tutors

Rodolfo
Certified Tutor

The University of Texas at San Antonio, Bachelor of Science, Mathematics.

View Calculus Tutors

Maya
Certified Tutor

York University, Bachelor of Science, Computer Science.

View Calculus Tutors

Blossom
Certified Tutor

University of Maryland-Baltimore County, Bachelor of Science, Computer Science.

All Calculus 1 Resources

10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

MCAT Tutors in Philadelphia, English Tutors in Seattle, Algebra Tutors in Chicago, Spanish Tutors in Miami, Biology Tutors in Phoenix, MCAT Tutors in Houston, Spanish Tutors in Chicago, Biology Tutors in Philadelphia, ACT Tutors in New York City, Spanish Tutors in Atlanta

Popular Courses & Classes

GRE Courses & Classes in Boston, GRE Courses & Classes in Los Angeles, GMAT Courses & Classes in Washington DC, ACT Courses & Classes in Phoenix, SSAT Courses & Classes in San Francisco-Bay Area, SAT Courses & Classes in New York City, GRE Courses & Classes in Washington DC, SAT Courses & Classes in Washington DC, LSAT Courses & Classes in Washington DC, SSAT Courses & Classes in Seattle

Popular Test Prep

GMAT Test Prep in Philadelphia, SAT Test Prep in Dallas Fort Worth, GMAT Test Prep in Washington DC, LSAT Test Prep in Houston, GRE Test Prep in San Francisco-Bay Area, SAT Test Prep in Los Angeles, SSAT Test Prep in Dallas Fort Worth, GMAT Test Prep in Dallas Fort Worth, ACT Test Prep in Denver, 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 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #29 : How To Find Area Of A Region

Example Question #2991 : Functions

Example Question #4021 : Calculus

Example Question #32 : How To Find Area Of A Region

Example Question #31 : Area

Example Question #32 : Area

Example Question #34 : How To Find Area Of A Region

Example Question #111 : Regions

Example Question #35 : How To Find Area Of A Region

Example Question #36 : How To Find Area Of A Region

All Calculus 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: