Application of Integrals : Volumes

Download Maple Worksheet

We all know that the volume of a right circular cylinder with radius r and height h is .

 

In general, a solid is called a cylinder (or right cylinder), as illustrated below, is bounded by a plane region B , called the base , and a congruent region B' in a parallel plane.

¡@

If the area of region B is A , then the cylinder has volume A h.

If we consider the cylinder with base {  ( ) | , is in []  } and height h as illustrated below :

We know that the area of the base is , so the volume of the solid is .

 

For a solid S that is not a cylinder we first "cut" S into pieces and approximate each piece by a cylinder. For instance, let's consider the solid illustrated below, which is obtained rotating the region bounded by and about the z -axis.

We first "cut" the solid into n thin pieces. ( Click on the animation to see how it goes.)

Since each horizontal cross-section ( the intersection of the solid and a plane) is a circle, we can approximate each piece by a circular cylinder.

n = 10 :

n = 20 :

 

Consider the piece cut by the horizontal plane with height and . From the graph above, we see that the radius r of the cross-section at height z is given by ( since ).

 

So we can approximate the piece by .   In this way, we get the total volume of all the pieces is given by

 

The approximation become better and better as .(  that is, the slices get thinner and thinner )

Therefore the volume of the solid is .

 

 

Let S be the solid that lies between and . If the cross-sectional area of S in the plane , through and perpendicular to the x -axis, is A( x ), where A is a continuous function, then the volume of S is

 

Let's find the volume of a pyramid whose base is a square with side 2 and whose height is 2.

As we can see from the graph above, each cross-section of the pyramid obtained by intersecting the plane

()

is a square whose side a satisfies the equation

Hence the area of the cross-sectional area in is and the volume of the pyramid is

.

Can you find the volume of a pyramid whose base is a square with side a and whose height is h ?

 

A wedge is cut out of a circular cylinder of radius 4 by two planes. One plane is perpendicular to the axis of the cylinder. The other intersects the first at an angle of 30 degrees along a diameter of the cylinder. What is the volume of the wedge ?

In the graphs above, we see that a cross-section perpendicular to the x -axis at distance x from the origin is a triangle whose base is and whose height is .

Thus, the cross-sectional area is and the volume is

= = .

 

 

Now let's the solid obtained by rotation the region enclosed by the curvesandabout the x -axis.  What is the volume of the solid ?

¡@

Here are animations of showing how we can get the solid :

1. The surface obtained by rotating the curve about the x -axis.

2. The surface obtained by rotating the curve about the x -axis.

3. Rotating the region enclosed by the curves and about the x -axis.

Notice the the curves intersect each other at ( ) and ( ).

Here is a picture of the solid.

Here is an animation of the cross-section.

In the animation above, we see that the cross-section in the plane is an annular ring with inner radius and outer radius , so the cross-sectional area is

Therefore, the volume of the solid is

.

 

If the solid is obtained by rotation the region enclosed by the curves and about . What is the volume of the solid ?

 

Here is a picture of the solid.

In the animations above, we see that the cross-section in the plane is an annular ring with inner radius and outer radius , so the cross-sectional area is

Therefore, the volume of the solid is

.