Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 1 : Regions

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 #101 : Area

Find the area of the region bounded by the function $\small f(x)=4-4e^{-2x}$ and the -axis over the interval $\small x=[0,2]$ .

Possible Answers:

$\small 8+2e^{-4}$

$\small 10-2e^{-4}$

$\small 10+2e^{-4}$

$\small 8-2e^{-4}$

$\small 6+2e^{-4}$

Correct answer:

$\small 6+2e^{-4}$

Explanation:

To find the area of the region between $\small f(x)=4-4e^{-2x}$ and the x-axis, integrate the function over the specified interval:

For this particular function use the following rules to integrate.

$\int x^n=\frac{x^{n+1}}{n+1}$ and $\int e^u=\frac{1}{\frac{du}{dx}}e^u$

Therefore we get,

$\small A=\int_0^2(4-4e^{-2x})dx$

$\small A=4x+2e^{-2x}|_0^2$

$\small A=(8+2e^{-4})-(4(0)+2e^{-2(0)})$

$\small A=8+2e^{-4}-2$

$\small \small A=6+2e^{-4}$ .

Report an Error

Example Question #102 : How To Find Area Of A Region

A construction company is pouring concrete pads for an abstract art-themed promenade on a college campus. One slab's overhead view is given by region contained between the functions f(x)=x+2 and $g(x)=x^{2}-4$ and to the right of the -axis, where gives the size in feet. If the slab is two feet deep, what is the volume occupied by the concrete slab?

Possible Answers:

$\frac{68}{3}ft^{3}$

$\frac{27}{2}ft^{3}$

$54 ft^{3}$

$27 ft^{3}$

Correct answer:

$27 ft^{3}$

Explanation:

Since the depth of the slab is constant, we know it's volume is given by:

V= Ah =2A

The area is, of course given by an integral. To calculate it, we must know where the two functions intersect:

$x+2 = (x^{2}-4)$

$x^{2}-x-6 = 0$

Which has roots x=-2 and x=3 . Since we are concerned with area to the right of the y-axis, we discard the negative root. So the area is given by:

$A=\int_{0}^{3}x+2-(x^{2}-4)dx$

$\int_{0}^{3}-x^{2}+x-6dx$

$[-\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+6x]_{0}^{3}$

= $\frac{27}{2}ft^{2}$

So $V = 2ft(\frac{27}{2}ft^{2})= 27ft^{3}$

Report an Error

Example Question #101 : How To Find Area Of A Region

Find the dot product of the two vectors.

$<9,6,4>\cdot<3,1,6>$

Possible Answers:

None of these

Correct answer:

Explanation:

The dot product of two vectors can be found by multiplying the first element of one by the first element in the other and adding it to the product of the second elements of each and so on. Using this method the dot product becomes

$<9,6,4>\cdot<3,1,6>=9\cdot3+6\cdot1+4\cdot6$

=27+6+24=57

Report an Error

Example Question #101 : How To Find Area Of A Region

Find the dot product of <2,1,2> and <5,5,5>

Possible Answers:

None of these

Correct answer:

Explanation:

$<2,1,2>\cdot<5,5,5>=2\cdot5+1\cdot5+2\cdot5$

=10+5+10=25

Report an Error

Example Question #105 : How To Find Area Of A Region

The following problem asks for the development of a maximization problem.

With 64 inches of string, you are supposed to find the most effective use of your yarn by creating a four-sided object with the most area possible. This can be solved using basic calculus, which of the following shows the correct two equations necessary to solve this equation?

( and represent different sides of the object while represents the surface area inside the object)

Possible Answers:

2a+2b=A

ab=64

a+b=A

ab=64

4ab=A

2a+2b=64

ab=A

a+b=64

ab=A

Correct answer:

2a+2b=64

ab=A

Explanation:

If we look at the object, we will notice that we have four sides that are added up to give us our total length of string which is . This can be expressed as 2a+2b=64 . We also know that if we multiply the sides and , we will find the area, which we are trying to maximize. This leads us with our answer, and ab=A . Can you find the length of each side?

Report an Error

Example Question #106 : How To Find Area Of A Region

Find the area bounded by $y = \sqrt{x}$ and y = x^3 on the interval [0,1] .

Possible Answers:

$\frac{7}{12}$

$\frac{2}{3}$

$\frac{5}{12}$

$\frac{1}{4}$

Correct answer:

$\frac{5}{12}$

Explanation:

To find the area between these two curves, we must start by determining if one function is greater than the other function over the entire interval. Because $\sqrt{x} > x^3$ over the interval [0,1] , we can simply subtract the area under y = x^3 from the area under $y = \sqrt{x}$ . Set up the integration as follows:

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

To evaluate this integral, follow the power rule for integrals:

$\int x^n = \frac{x^{n+1}}{n+1}$

Realizing that $\sqrt x = x^{\frac{1}{2}}$ , the integral evaluates to:

$(\frac{2\cdot 1^{3/2}}{3} - \frac{1^4}{4}) - (\frac{2\cdot0^{3/2}}{3} - \frac{0^4}{4})$

$\frac{2}{3} - \frac{1}{4}$

$\frac{8}{12} - \frac{3}{12}$

$\frac{5}{12}$

Report an Error

Example Question #4091 : Calculus

Find the area underneath the curve to the x-axis of the function on the interval [0,15]

f(x)=50 .

Possible Answers:

750 square units

600 square units

1000 square units

500 square units

Correct answer:

750 square units

Explanation:

In order to find the area underneath the curve to the x-axis on the interval [a,b] we must solve the integral

$\int_{a}^{b} \left | f(x)\right |\, dx$ .

Because the function is always positive on the interval [0,15] we solve the integral

$\int_{0}^{15} 50 \, dx$ .

When taking the integral we apply the inverse power rule which states

$\int x^n = \frac{x^{n+1}}{n+1}$ .

As such

$\int_{0}^{15} 50 \, dx = 50x|_0^{15}$ .

And by the corollary of the first Fundamental Theorem of Calculus

$50x|_0^{15}=50*15-50*0=750$ .

As such the area is

750 square units.

Report an Error

Example Question #102 : How To Find Area Of A Region

Find the area underneath the curve to the x-axis of the function on the interval [0,2]

f(x)=x^4 .

Possible Answers:

square units

$\frac{32}{5}$ square units

$\frac{15}{2}$ square units

square units

Correct answer:

$\frac{32}{5}$ square units

Explanation:

In order to find the area underneath the curve to the x-axis on the interval [a,b] we must solve the integral

$\int_{a}^{b} \left | f(x)\right |\, dx$ .

Because the function is always positive on the interval [0,2] we solve the integral

$\int_{0}^{15} x^4 \, dx$ .

When taking the integral we apply the inverse power rule which states

$\int x^n = \frac{x^{n+1}}{n+1}$ .

As such

$\int_{0}^{2} x^4 \, dx = \frac{x^5}{5}|_0^{2}$ .

And by the corollary of the first Fundamental Theorem of Calculus

$\frac{x^5}{5}|_0^{2}=\frac{2^5}{5}-0=\frac{32}{5}$ .

As such the area is

$\frac{32}{5}$ square units.

Report an Error

Example Question #101 : How To Find Area Of A Region

Find the area underneath the curve to the x-axis of the function on the interval [0,1]

f(x)=e^x .

Possible Answers:

square units

e-1 square units

square units

square unit

Correct answer:

e-1 square units

Explanation:

In order to find the area underneath the curve to the x-axis on the interval [a,b] we must solve the integral

$\int_{a}^{b} \left | f(x)\right |\, dx$ .

Because the function is always positive on the interval [0,1] we solve the integral

$\int_{0}^{1} e^x \, dx$ .

Because the antiderivative of the exponential function is the exponential function itself, we obtain

$\int_{0}^{1} e^x \, dx = e^x|_0^1$ .

And by the corollary of the first Fundamental Theorem of Calculus

e^x|_0^1 = e^1-e^0=e-1 .

As such the area is

e-1 square units.

Report an Error

Example Question #101 : How To Find Area Of A Region

Evaluate the following integral to find the area of the region bound by the function and the given limits.

$\int_{0}^{5}x^4-6x^2+8x+6dx$

Possible Answers:

505 units^2

5 units^2

275 units^2

475 units^2

Correct answer:

505 units^2

Explanation:

Evaluate the following integral to find the area of the region bound by the function and the given limits.

$\int_{0}^{5}x^4-6x^2+8x+6dx$

Begin by recalling the integration rule for polynomials. All we need to do to integrate a term of a polynomial is raise the exponent by 1 and divide by the new number.

So, we go from this

$\int_{0}^{5}x^4-6x^2+8x+6dx$

To this

$\frac{x^5}{5}-\frac{6x^3}{3}+\frac{8x^2}{2}+6x+c|_0^5$

We now have successfully integrate our function, but we still must evaluate it with the given limits. To do so, simply plug in our lower and upper limits and find the difference between them. Note, this is step is simplified by the fact that our lower limit is 0, because plugging zero into our function will just yield "c"

So...

$\frac{5^5}{5}-\frac{6(5)^3}{3}+\frac{8(5)^2}{2}+6(5)+c-c$

5^4-2*5^3+4*5^2+30=505

Making our answer

505 units^2

Report an Error

View Calculus Tutors

Jesus Ricardo
Certified Tutor

Benemerita Universidad Autónoma de Puebla, Bachelor, Physics.

View Calculus Tutors

Mayowa
Certified Tutor

Logan University, Bachelor of Science, Biology, General.

View Calculus Tutors

Luis
Certified Tutor

State University of New York at New Paltz, Doctor of Science, Health Physics.

All Calculus 1 Resources

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

Popular Subjects

Physics Tutors in San Francisco-Bay Area, SSAT Tutors in Chicago, GMAT Tutors in Boston, SSAT Tutors in San Diego, LSAT Tutors in Miami, Statistics Tutors in Chicago, GMAT Tutors in Phoenix, LSAT Tutors in Atlanta, Spanish Tutors in Washington DC, MCAT Tutors in Boston

Popular Courses & Classes

GRE Courses & Classes in San Francisco-Bay Area, GMAT Courses & Classes in New York City, MCAT Courses & Classes in Miami, GMAT Courses & Classes in San Diego, MCAT Courses & Classes in Los Angeles, MCAT Courses & Classes in Phoenix, GRE Courses & Classes in Denver, SSAT Courses & Classes in Chicago, SSAT Courses & Classes in Washington DC, LSAT Courses & Classes in Philadelphia

Popular Test Prep

GMAT Test Prep in New York City, MCAT Test Prep in Chicago, SSAT Test Prep in Chicago, MCAT Test Prep in Denver, ISEE Test Prep in Chicago, MCAT Test Prep in Boston, GRE Test Prep in Denver, GMAT Test Prep in Phoenix, ACT Test Prep in New York City, ISEE Test Prep in Atlanta

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 : Regions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #101 : Area

Example Question #102 : How To Find Area Of A Region

Example Question #101 : How To Find Area Of A Region

Example Question #101 : How To Find Area Of A Region

Example Question #105 : How To Find Area Of A Region

Example Question #106 : How To Find Area Of A Region

Example Question #4091 : Calculus

Example Question #102 : How To Find Area Of A Region

Example Question #101 : How To Find Area Of A Region

Example Question #101 : 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: