Volumes by Slicing and Rotation About an Axis

In summary, the problem involves finding the volume of a solid with a base in the shape of a disk and cross-sections of isosceles right triangles. The base of the triangle is 2y, which can be derived from the equation of the circle. The element of volume is found using the thickness of the cross-section slab and the volume can be found by integrating the equation in terms of x. The final answer for the volume is 8/3.
  • #1
TheTaoOfBill
6
0

Homework Statement


I'm having issues with this section in calc. I'm not at all sure what I'm doing!

Here is the problem I'm having trouble with:

Directions: Find the volume of the solids.

Problem: The base of the solid is the disk X^2 + Y^2 <= 1. The cross-sections by planes perpendicular to the Y-axis between Y= -1 and Y= 1 are isosceles right triangles with one leg in the disk.

So essentially it shows a right isosceles triangle where right angle touches the left edge of the circle and the base of the right triangle extends to the other side of the circle.



Homework Equations



about all I really understand what to do is find the area of the triangle.

A(x) = 1/2 BH
B=2x=H
1/2(2x)(2x) = 2x^2

The Attempt at a Solution



I honestly don't have a clue what I'm doing in this section. But if I were to guess I'd take the integral of the area from 1to-1 since that's my interpretation of what the book says.

So V = int from 1to-1(2x^2) = 4/3

But that doesn't sound right at all.

It also talks about the disk and washer methods in this section but I'm not sure if those apply. Any help would be appreciated.
 
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  • #2
You have the idea, but the plane cross sections are perpendicular to the y axis, not parallel to it. So the base of the triangle is 2y, which you can get in terms of x from the equation of the circle. Your cross section slab has thickness dx. So your element of volume is

dV = (1/2)(2y)2 dx

Get it all in terms of x and go.
 
  • #3
LCKurtz said:
You have the idea, but the plane cross sections are perpendicular to the y axis, not parallel to it. So the base of the triangle is 2y, which you can get in terms of x from the equation of the circle. Your cross section slab has thickness dx. So your element of volume is

dV = (1/2)(2y)2 dx

Get it all in terms of x and go.

ok so
dV = 1/2(2(sqrt(1-x^2))

so V = int from -1 to 1(1/2(2(sqrt(1-x^2))) = pi/2 ?
 
  • #4
Did you miss a squared term?
 
  • #5
LCKurtz said:
Did you miss a squared term?

Ah.. right...

So..

V = int from -1 to 1(1/2(2(sqrt(1-x^2))^2) = 8/3
 

FAQ: Volumes by Slicing and Rotation About an Axis

1. What is meant by "volume by slicing and rotation about an axis?"

Volume by slicing and rotation about an axis is a method used to calculate the volume of a solid figure by breaking it down into smaller, simpler shapes and rotating them around a given axis. This technique is commonly used in calculus and geometry to find the volume of objects with irregular or curved shapes.

2. How do you find the volume using this method?

To find the volume using volume by slicing and rotation, you first need to identify the axis of rotation and the shape that will be rotated around it. Then, you divide the shape into infinitely thin slices and integrate the cross-sectional areas of each slice along the axis of rotation. Finally, you add up all of these integrals to get the total volume.

3. Can this method be used for any shape?

No, this method is most commonly used for finding the volume of objects with circular or cylindrical shapes. It can also be used for other shapes, such as spheres or cones, as long as they can be broken down into infinitely thin slices and rotated around an axis.

4. What is the difference between volume by slicing and volume by rotation?

Volume by slicing involves dividing the shape into thin slices parallel to the axis of rotation, while volume by rotation involves dividing the shape into thin slices perpendicular to the axis of rotation. The method used depends on the shape of the object and the axis of rotation.

5. What are some practical applications of volume by slicing and rotation?

This method has many practical applications in fields such as engineering, physics, and architecture. It can be used to calculate the volume of complex objects, such as aircraft wings or suspension bridges, as well as to determine the volume of irregularly shaped objects in medical imaging or fluid dynamics.

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