1. Volumes of Solids Solids of Revolution Approximating Volumes
Volumes as Integrals of Areas of Slices
Examples
General Solids of Revolution
Other Types of Solids
Long Domains – Improper Solids

2. Solids of revolution
Volumes of such solids of revolution can be computed by definite integrals in the same way as one computes areas of domains by integrals.
Mika Seppälä: Volumes

3. Approximating Volumes
The picture on the right shows a left rule approximation of the domain bounded by the function f. Letting the green boxes rotate around the x-axis one gets an approximation of the solid of revolution by cylinders.
The denser the (left rule) approximation of the domain is, the better the approximation of the solid (by cylinders) is.
Solid on the left, approximation on the right.
Mika Seppälä: Volumes

4. Riemann Sums for Volumes
Definition
Mika Seppälä: Volumes

5. Volumes as Integrals of Areas of Slices
The slice is obtained by slicing the solid by a plane perpendicular to the x-axis and intersecting the axis at the point x. The slice is a disk of radius f(x).
In the picture on the right, the red curve is the graph of the function f, x-axis is tilted down and y-axis is tilted to the right.
Mika Seppälä: Volumes

6. Volume of a Ball
Example
Mika Seppälä: Volumes

7. Volume of a Torus (1) Example
The volume of the torus thus formed has to be computed in two steps.
Mika Seppälä: Volumes

8. Volume of a Torus (2)
Mika Seppälä: Volumes

9. Volume of a Torus (3)
Mika Seppälä: Volumes

10. General Solids of Revolution
In general, a solid of revolution is obtained whenever a domain rotates around some line. The same methods as before can be applied also in such cases, but computations may become more complicated.
As an example consider a solid of revolution obtained by letting the domain bounded by the line y=x and the parabola y=x2 rotate around the line y=x.
Mika Seppälä: Volumes

11. General Solids of Revolution (2)
Mika Seppälä: Volumes

12. Other Types of Solids
Volumes of other types of solids can also be computed by integration. The idea is to perform the following:
Slice the solids vertically along some straight line going through the solid.
Express the area of the slice as a function of the intersection point of the line and the slice.
Integrate this area function to find the volume of the solid.
An example of this type of solid is the “hat” above. It is a solid whose bottom face is a disk with radius 1 and center at the origin, and whose every cross section perpendicular to the x-axis is a square. The volume of this solid can be computed by suitable slicing of the solid.
Mika Seppälä: Volumes

13. Volume of the Hat
Mika Seppälä: Volumes

14. Long Domains
Consider the function f(x)=1/(1+x+sin(x)) in the interval [1, ∞ ]. The graph of this function is shown on the right. An application of the Comparison Theorem shows that the area, extending to the infinity, under this graph is infinite.
These figures are not in scale.
Now let this infinite domain rotate around the x-axis. One gets a solid of revolution as shown above.
Mika Seppälä: Volumes

15. Long Domains (2)
This is an example of a domain with infinite area which produces a finite volume solid of revolution.
Mika Seppälä: Volumes