## Understanding Calculus

#### e-Book for \$4

 Preface 1. Why Study   Calculus 2. Numbers 3. Functions 4. The Derivative 5. Differentiation 6. Applications 7. Free Falling   Motion 8. Understanding  Derivative 9. Derivative  Approximations 10. Integration   Theory 11. Understanding   Integration 12. Differentials

 Inverse Functions Exponents Exponential  Functions Applications of   Exponential  Functions Sine and Cosine   Function Sine Function Sine Function -   Differentiation and   Integration Oscillatory Motion Mean Value   Theorem Taylor Series More Taylor Series Integration   Techniques

+ Share page with friends
Friend Emails:
Your Email - optional:
CHAPTER 11

### Chapter 11 - Understanding Integration

#### Section 11.2 - Geometric Applications

Let us consider some geometric examples of integration that are easy to visualize and understand. Following geometric applications we will move onto more physical applications from science and engineering.

The first example we will study is that for finding the area of a shape whose height is changing with its length. By definition the area of a rectangle is given by the equation:

What if the height of a particular shape were changing with respect to its length, expressed by the function, ? The graph of the function and shape would look like this:

What is peculiar about the shaded area? The only apparent characteristic is that its height is changing with the length such that as the length increases the height increases.

Therefore the area of the object is not a simple matter of:

In the graph, height is a function of length. The area equation for the shape now becomes:

Over an infinitely small length, dl, the area of one rectangle has constant height, and length, dl . Over an infinitely small interval, the area under the curve changes by the area of one rectangle. This can be expressed as:

The infinite sum of the inscribed rectangles over an interval of l=a to l=b corresponds exactly to the net change in the area of the shape, or the area under the curve from l=a to l=b. Thus the area of the shape from l=1 to l=3 is:

Evaluating the integral:

The total area is then 20,or looking at the graph of

It is important to understand, that the function gives us the net change in the area defined by a changing height, as l changes from some value a to b. Since then in this unique case will give us the area the shape for any length, l. However keep in mind that we are calculating how much a situation changes, as its defining conditions change.

Now let us look at volumes. Let us say we take the same shape with length, l, and height, and revolve it around the x-axis. How do we find the volume of the conical cylinder generated?

All we have is a cylinder of varying or changing radius. The radius is give as a function of length by;

The equation for the volume of a cylinder is:

As the length goes from some value, a to b the radius changes with respect to the length. The equation can only hold true over an infinitely small interval, dl. The change in the volume of one infinitely small cylinder with length, dl, is

The net change in the volume of the cylinder as the length changes from 0 to 6 is found by integrating the differential,

Next section -> Section 11.3 - The Systematic Approach to Integration

 Type in any questions, corrections or suggestions for this page here. Email: