Hippocampus Calculus: Properties of the definite integral

Previously, we defined the definite integral as the limit of a Riemann sum. We assumed that b is greater than a.

However, suppose a is greater than b.

We see that the integral from b to a is the negative of the integral from a to b. This is the first of several properties of definite integrals that we will look at in this lesson.

Now, let's look at the area under the curve in this graph. Use the slider to vary the value of b.

Suppose we have the definite integral of some constant c.

When the limits of integration are equal, the value of the definite integral is zero.

The area under the curve is that of a rectangle, as shown in this graph.

The definite integral of a constant is equal to the constant times the difference between the limits of integration.

Now, consider this graph of a function f (x). Let's plot another function that is three times f (x).

Consider this graph of the two functions f (x) and g (x).

Now, use the slider to change the area under the curve of g (x). Notice how the three areas compare.

The area under the curve of the combined functions is equal to the sum of the areas under each function curve.

This illustrates the sum property of definite integrals. The integral of a sum is equal to the sum of the integrals.

Now, what about the integral of a difference of two functions?

We can change it into an integral of a sum and use the sum property to show it as two integrals.

We then use the constant times a function property to bring the negative sign outside the second integral.

We then see that the integral of a difference is the difference of the integrals.

Now, consider the graph of a curve on the interval [a, b].

Suppose we have another point, c, that lies between a and b.

Use the slider to vary the value of c. Notice how the area under the curve from a to b compares to the area under the curve from a to c, and c to b.

This shows how we can combine integrals of the same function over adjacent intervals into a single integral over the entire interval.

Now, let's look at an example. Given the value of these definite integrals, let's evaluate the following definite integral.

First, we use the difference rule to break it into two integrals.

We can determine the value of the first integral using the constant rule.

By reversing the limits of integration, we can determine the value of the second integral from that of the known integral.

We then combine our two results to arrive at the value of the original definite integral.

There are some other useful properties of definite integrals for us to consider. Here we have the graph of an even function.

The areas under the curve from -a to zero, and from zero to a, can be determined from the given definite integrals.

For an even function, the integral from -a to a is equal to twice the integral from zero to a.

The definite integral of an odd function from -a to a is equal to zero.

Now, suppose we know the value of this definite integral. We also know that the function f (x) is even.

Suppose that f (x) is greater than or equal to zero, as illustrated in this graph.

Consider the graph of the two functions f (x) and g (x), where f (x) is greater than or equal to g (x).

If f (x) is greater than or equal to g (x), then the integral of f (x) is greater than or equal to the integral of g (x) on the same interval.

Now, consider this graph of some function f (x).

The maximum value on the interval [a, b] is denoted by fmax, and the minimum value on the same interval is denoted by fmin.

Look at the area of the rectangle defined with a height of fmin and a width of b minus a.

Now, look at the area of the rectangle defined with a height of fmax and a width of b minus a.

It follows that, on a given interval, the definite integral that represents the area under the curve is greater than or equal to the area of the rectangle defined with a height of fmin, and is less than or equal to the area of the rectangle defined with a height of fmax.

Use the slider to increase or decrease the value of g (x).

The area under the curve is always positive or zero.

Therefore, the definite integral that represents this area is always greater than or equal to zero.

Now, let's look at some comparison properties of definite integrals. Each of these assumes that the variable x is greater than or equal to a, and less than or equal to b.

Now, consider the graph of an odd function.

Let's plot a third curve that represents the sum of the two functions.

The area under each curve is displayed in the text boxes.

Next, let's approximate the area under each curve using Reimann sums.

We see that the height of a rectangle under the second curve is three times that of the corresponding rectangle under the first curve. Therefore, the area under the second curve is three times that under the first curve.

This illustrates another property of definite integrals. The integral of a constant times a function is equal to the constant times the integral of the function.

How do we define Dx in this case? Click the "Submit" button after selecting your answer.

What is the area under the curve when b equals a? Click the "Submit" button after selecting your answer.

What is the area of this rectangle? Click the "Submit" button after selecting your answer.

How does the area under the second curve compare to the area under the first curve? Click the "Submit" button after selecting your answer.

What relationship is shown by the areas under the three curves? Click the "Submit" button after selecting your answer.

What relationship is shown by the three areas? Click the "Submit" button after selecting your answer.

What is the value of the first integral? Click the "Submit" button after entering your answer.

Now, what is the value of the second integral? Click the "Submit" button after entering your answer.

How do you think these two areas compare? Click the "Submit" button after selecting your answer.

What is the value of the definite integral on the interval [-a, a]? Click the "Submit" button after selecting your answer.

What is the value of the following integral? Click the "Submit" button after entering your answer.

How does the area under the curve of f (x) compare to that of under the curve of g (x)? Click the "Submit" button after selecting your answer.

How does the area of this rectangle compare to the area under the function curve on the same interval? Click the "Submit" button after selecting your answer.