Find the volume of a cylinder of radius r, Mathematics

Assignment Help:

Find the volume of a cylinder of radius r and height h.

Solution : Here, as we mentioned before starting this illustration we actually don't require using an integral to get this volume, but it is a good illustration to exemplify the method we'll require to use for these kinds of problems.

We will start off with the diagram of the cylinder below.

84_More Volume Problems 1.png

We will center the cylinder on the x-axis and the cylinder will begin at x = 0 and end at x = h as demonstrated. Remember that we are only choosing this exact set up to find an integral in terms of x and to create the limits nice to deal along with. There are various other orientations which we could use.

What we require now is to find a formula for the cross-sectional area at any x. During this case the cross- sectional area is constant and it will be a disk of radius r. Thus, for any x we'll have the subsequent cross-sectional area,

A (x)= pr2

After that the limits for the integral will be as 0 ≤ x ≤ h as i.e. the range of x wherein the cylinder lives. Now there is the integral for the volume,

1236_More Volume Problems 2.png

Therefore, we find the expected formula.

And, recall we are using r to classify the radius of the cylinder. Whereas r can clearly take various values this will never change once we begin the problem. Cylinder's radius does not change in the middle of a problem and therefore as we move along the center of the cylinder that is the x- axis, r is a fixed number and was not change. Conversely, this is a constant which will not change when we change the x. Thus, as we integrated with respect to x the r will be a constant as much as the integral is associated. The r can after that be pulled out of the integral as demonstrated, though that's not needed, we just did this to make the point.  At this point we are only integrating dx and we identify how to do that.

While we evaluate the integral keep in mind that the restrictions are x values and therefore we plug in the x and NOT the r.  Again, keep in mind that r is only a letter which is being used to represent the radius of the cylinder and, once we start the integration, is assumed to be a fixed constant.

Since observed before we started this illustration if you are having trouble along with the r just think of what you would do whether there was a 2 there in place of an r. In this problem, as we're integrating with respect to x, both the 2 and the r will behave in similar way. Note though that you must NEVER really replace the r with a 2 as that WILL guide to a wrong answer.  You must just think of what you'd do IF the r was 2.

Therefore, to work these problems we will first require finding a sketch of the solid along with a set of x and y axes to assist us notice what's going on. At the extremely least we will require the sketch to find the limits of the integral, but we will frequently require this to see just what the cross-sectional area really is. Once we have the draw we'll require to find out a formula for the cross-sectional area and after that do the integral.


Related Discussions:- Find the volume of a cylinder of radius r

Domain and range, Taxable income Tax rate 0 - $18,200 0% $18,201- $37,000 1...

Taxable income Tax rate 0 - $18,200 0% $18,201- $37,000 19% $37,001 - $80,000 32.5% $80,001- $180,000 37% $180,001 and over 45% if this is graphed as a step fuction graph whats t

Find third order partial derivatives, Question: Find all third order pa...

Question: Find all third order partial derivatives for the function   F(x,y)= log xy+ e (x+y) -x/y.

Calculus three, i would like answers to these questions i will give you as ...

i would like answers to these questions i will give you as soon as possible

Important points about the alternating series test, Important Points About ...

Important Points About the Alternating Series Test There are a several things to note about this test.  Very first, unlike the Integral Test and the Comparison or Limit Compari

Advantages of peer interaction in learning maths, Can you think of some mor...

Can you think of some more advantages of peer interaction and child-to child learning? If you agree that children learn a lot from each other, then how can we maximise such oppo

Define a complete lattice, Define a complete lattice and give one example. ...

Define a complete lattice and give one example. Ans:  A lattice (L, ≤) is said to be a complete lattice if, and only if every non-empty subset S of L has a greatest lower bound

Linear equation, develop any two linear equation which are reducible into l...

develop any two linear equation which are reducible into linear form from our daily life by cross multiplication

I need help with math, can i get help with math just with fractions i want ...

can i get help with math just with fractions i want to catch up with my class

Example of hcf, Example  Find the Highest Common Factor of 54, 72...

Example  Find the Highest Common Factor of 54, 72 and 150. First we consider 54 and 72. The HCF for these two quantities is calculated as follows:

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd