Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

varsity tutors app store varsity tutors android store

Example Questions

Example Question #61 : Volume

Using the Shell Method, find the volume of the solid obtained by rotating the region bound by  and  about the line 

 

Possible Answers:

Correct answer:

Explanation:

The parabola's x-intercepts are 0 and 1. The radius is , the circumference is  and the height is .

 

Integrating, we get:

Example Question #3981 : Calculus

Find the volume of the solid obtained by rotating the region bound by   and  about the y-axis using the Shell Method. 



Possible Answers:

Correct answer:

Explanation:

The radius of a single shell can be expressed as , the circumference can be expressed as , and the height can be expressed as 

 

To obtain the volume of all the shells, we integrate the overall volume.  Our bounds can be found by calculating the endpoints, or where . Thus, our endpoints are at  and .

Integrating, we get: 

Example Question #3981 : Calculus

Find the volume of the solid obtained by rotating the region bound by   and  about the y-axis using the Shell Method. 



Possible Answers:

Correct answer:

Explanation:

The radius of a single shell can be expressed as , the circumference can be expressed as , and the height can be expressed as 

 

To obtain the volume of all the shells, we integrate the overall volume.  Our bounds can be found by calculating the endpoints, or where . Thus, our endpoints are at  and .

Integrating, we get: 

Example Question #71 : How To Find Volume Of A Region

Use the Shell Method to find the volume of the solid obtained by rotating the region under the curve  from  to  about the x-axis

.

Possible Answers:

Correct answer:

Explanation:

To use shells we relabel the curve  as .

The radius of a single shell can be expressed as , the circumference can be expressed as , and the height can be expressed as 

 

To obtain the volume of all the shells, we integrate the overall volume.  Our bounds are given as  and .

Integrating, we get: 

 

Example Question #71 : How To Find Volume Of A Region

Use the Shell Method to find the volume of the solid obtained by rotating the region under the curve  from  to  about the x-axis

.

Possible Answers:

Correct answer:

Explanation:

To use shells we relabel the curve  as .

The radius of a single shell can be expressed as , the circumference can be expressed as , and the height can be expressed as 

 

To obtain the volume of all the shells, we integrate the overall volume.  Our bounds are given as  and .

Integrating, we get: 

 

Example Question #3982 : Calculus

Using the method of cross sections, find the volume of the solid bounded by the function

and the x-axis.

The cross sections are squares perpendicular to the x-axis.

Possible Answers:

Correct answer:

Explanation:

The formula for the volume of a solid using the method of cross sections perpendicular to the x-axis on the interval  is 

Where  is the area of the cross section.

Because the cross sections are squares with a side length of    the formula becomes

In order to solve the integral we use the inverse power rule which says

  

Applying this rule we get

And by the corollary of the Fundamental Theorem of Calculus we get

Example Question #71 : Volume

Determine the volume of the solid obtained by rotating the region bounded by  and  about the line .

Possible Answers:

Correct answer:

Explanation:

To find the endpoints, we set:

and thus, we get y=4 and y=1 as our bounds.

The area can be found by:

Integrating over our volume, we get:

Example Question #71 : Volume

Determine the volume of the solid obtaining by rotating the region bounded by  and  about the line .

Possible Answers:

Correct answer:

Explanation:

To find the bounds, we set . Solving, we get , giving us bounds of  and .

 

The cross sectional area is then:

Integrating, we get the volume:

Example Question #71 : Regions

Find the volume of the solid fo revolution formed by rotating the region formed by the x-axis and the graph of  from x=0 to x=1, about the y-axis.

Possible Answers:

Correct answer:

Explanation:

Example Question #74 : Volume

The region enclosed by the curves  and  is rotated about the x-axis. Using discs or washers, find the volume of the resulting solid.

Possible Answers:

Correct answer:

Explanation:

The intersection points are (0,0) and (1,1). The cross-sectional area is calculated by finding the area between curves, using integration, getting:

The volume is therefore:

Learning Tools by Varsity Tutors