Interpretation of Integrals

Abstract

gives the volume of the 3-dimensional region
is the sum of infinitesimal quantities over all the points in
gives the mean value of the function on region

We are already familiar with the Riemann Sum and that interpretation of the integral. Or the volume under a scalar field for a double integral.

But this isn't very practical for applications. Instead, we should think of the integral as a tool for adding up an infinite number of infinitesimally small pieces.

Integrals with no Integrand

Example

Visualization

The Integral Doesn't Need to be Area

Intuition

The units come from both the integrand AND the infinitesimal.

Example

since .

Intuitively, we can say that which means we are adding up all the infinitesimal displacements ().

Example

The same intuition carries to multiple integrals.

Example

Mean Values

Definition

For numbers , the average of them is:

This is related to the definition of the definite integral

where the right side happens to be the mean value of on , and we denote this as

Definition

We can extend our definition to Multi-Integrals:

and

Example

Find the mean value of over

Example

With being the function representing the mass density of a surface in ,

Fundamental Theorem of Calculus (again)

fundamental theorem of calculus 1

Fundamental Theorem of Calculus, Part 1 (FTC 1)

Theorem

If is continuous on , then the function defined by

is differentiable on , and its derivative is

Or, more concisely

So we call the antiderivative of

Example

If

then