How Do You Determine the Height in Volume Calculations by Integration?

In summary, the attachement contains both problem and the attemp at a solution.Specifically, I am stuck on finding the equation for the hight for each solid.Should I just set up a random letter to indicate the heightor is there any way that I can figure out the equation for it?Link it somewhere like imageshack, take like a day to get your attachment approved!Can you see it now?take the 'O' as origin. Now, 'scan' through the x-axis.. like.. take a distance 'x' from the origin. From there, assume a cuboid of thickness 'dx' and find the area of that and then integrate within limits. The problem is that to find the width
  • #1
ada0713
45
0
http://img408.imageshack.us/my.php?image=picture20017hu6.jpg

The attachement contains both problem and the attemp at a solution.
Specifically, I am stuck on finding the equation for the hight for each solid.
(height "a" for both solids)
Should I just set up a random letter to indicate the height
or is there any way that I can figure out the equation for it?
 

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  • #2
Link it somewhere like imageshack, take like a day to get your attachment approved!
 
  • #3
Can you see it now?
 
  • #4
take the 'O' as origin. Now, 'scan' through the x-axis.. like.. take a distance 'x' from the origin. From there, assume a cuboid of thickness 'dx' and find the area of that and then integrate within limits. The problem is that to find the width and length of the cuboid, you're going to need the equation of the curve shown there.. which doesn't seem to be given or since the scan is small.. i can't see it.
 
  • #5
I think the curve should be a circle becuase the radius is indicated as "r" in the picture.
 
  • #6
I think he means the plane curve, as you will need to know how the slope rises and falls to know the volumes of your pieces.
 
  • #7
ada0713 said:
I think the curve should be a circle becuase the radius is indicated as "r" in the picture.

oh.. ur right mate.. should've figured that out :d

neways.. now that you have a circle, you can easily determine what the length of the cuboid will be at a distance of 'x' from the origin along the line. Also, the breadth is 'dx' and you can use ur image to get the height of the cuboid for a particular 'x'. [since the angle 'α' remains constant, you have a relation between 'x' and the height of the cuboid].

Find that and integrate within proper limits.. you should get the answer. Post here in case you have any problems.
 
  • #8
Okay.. I think I got the second one (with right triangle slice)
The "base" of the triangle should be sqrt((R^2-x^2)) (<- I got this from x^2+y^2=R^2)
and the "height" of the trangle is going to be sqrt((R^2-x^2))*tanα
(since tanα = height/base.. and you know "base" from above.. so you solve for height)

Than I pluged in two equations into
V=1/2*base*height*dx
and ended up with V= 1/2 (R^2-x^2)tanα dx

The rest of the integrating is not a big deal.. so i'll skip:)

Sorry that my work is kind of messy. I should have scanned my work
but I don't have a scanner with me at the moment..
 
Last edited:
  • #9
But the first one seems to be more complicated..
how do I find height of the rectangle?
 

Related to How Do You Determine the Height in Volume Calculations by Integration?

1. How do you find the volume using integration?

To find the volume using integration, you first need to set up a definite integral that represents the volume of the object. This involves choosing the correct variable to integrate with and setting appropriate limits of integration. Once the integral is set up, you can evaluate it using integration techniques such as the Fundamental Theorem of Calculus or integration by parts.

2. What objects can be measured using integration to find volume?

Integration can be used to find the volume of any object that has a defined boundary and can be represented as a mathematical function. This includes regular geometric shapes such as cubes, spheres, and cylinders, as well as irregular shapes such as cones, pyramids, and even more complex objects.

3. Can integration be used to find the volume of a 3D object with varying cross-sectional areas?

Yes, integration can be used to find the volume of a 3D object with varying cross-sectional areas. This is known as the method of cross-sections, where the object is split into infinitesimally small slices and the volume of each slice is added together using integration to find the total volume.

4. What is the difference between using integration and using the formula for calculating volume?

The formula for calculating volume is typically used for regular geometric shapes with known dimensions, while integration can be used for more complex and irregular shapes. Integration also allows for more accurate measurements as it takes into account any variations or changes in the shape of the object.

5. Are there any real-world applications of finding volume by integration?

Yes, there are many real-world applications of finding volume by integration. This can include calculating the volume of a chemical solution in a beaker, determining the volume of a medication in a syringe, or even measuring the volume of a lake or reservoir for water management purposes.

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