# 4. The Definite Integral

by M. Bourne

In the last section, we used the expression

int_a^bf(x)dx=[F(x)]_a^b=F(b)-F(a)

to find the area under a curve.

F(x) is the integral of f(x);

F(b) is the value of the integral at the upper limit, x = b; and

F(a) is the value of the integral at the lower limit, x = a.

This expression is called a definite integral. Note that it does not involve a constant of integration and it gives us a definite value (a number) at the end of the calculation.

We will use definite integrals to solve many practical problems. First, we see how to calculate definite integrals.

### Example 1

Evaluate int_1^5(3x^2+4x+1)dx

Note that our final answer is a number and does not involve "+ K". We are now dealing with definite integrals.

### Example 2

Evaluate int_4^9(2x+3sqrtx)dx

### Example 3

Evaluate int_1.2^1.6(5+6/x^4)dx

## Definite Integrals and Substitution

Recall the substitution formula for integration:

int u^n du=(u^(n+1))/(n+1)+K (if n ≠ -1)

When we substitute, we are changing the variable, so we cannot use the same upper and lower limits. We can either:

• Do the problem as an indefinite integral first, then use upper and lower limits later
• Do the problem throughout using the new variable and the new upper and lower limits
• Show the correct variable for the upper and lower limit during the substitution phase.

We will be using the third of these possibilities.

### Example 4

Find

int_-1^0x^3(1-2x^4)^3dx

using a substitution.

## Application: Work

Einstein riding his bicycle.

In physics, work is done when a force acting upon an object causes a displacement. (For example, riding a bicycle.)

If the force is not constant, we must use integration to find the work done.

We use

W=int_a^bF(x)dx

where F(x) is the variable force.

### Example 5

Find the work done if a force F(x)=sqrt(2x-1) is acting on an object and moves it from x = 1 to x = 5.

For more on work, see: 7. Work by a Variable Force.

## Application: Average Value

The average value of a function f(x) in the region x = a to x = b is given by:

"Average"=(int_a^bf(x)dx)/(b-a)

### Example 6

Find the average value of x(3x2 - 1)3 from 0 to 1.

Here is the graph of the situation:

## Application: Displacement

If we know the expression, v, for velocity in terms of t, the time, we can find the displacement (written s) of a moving object from time t = a to time t = b by integration, as follows:

s=int_a^bv\ dt

### Example 7

Find the displacement of an object from t = 2 to t = 3, if the velocity of the object at time t is given by

v=(t^2+1)/((t^3+3t)^2

See more on: displacement, velocity and acceleration as applications of integration.

NOTE 1: As you can see from the above applications of work, average value and displacement, the definite integral can be used to find more than just areas under curves.

NOTE 2: The definite integral only gives us an area when the whole of the curve is above the x-axis in the region from x = a to x = b. If this is not the case, we have to break it up into individual sections. See more at Area Under a Curve.

We now examine a definite integral that we cannot solve using substitution.

## Not every integral can be integrated using substitution...

Consider this question.

Find: int_0^1sqrt(x^2+1)\ dx

Attempted solution:

We try to put u = x^2+ 1.

Then we find the differential:

du = 2x\ dx

But the question does not have "2x\ dx" (it only has "dx") so we cannot replace anything in the question with "du" properly. This means we can't solve it using any of the integration methods used above. (Note: This question can be done using Trigonometric Substitution, however, but we don't meet trigonometric substitution until later.)

int_0^1sqrt(x^2+1)(2x)dx

This time, there is a "2x\ dx" involved in the question and so we would be able to let:

u = x^2+ 1

Then we would find the differential:

du = 2x\ dx

Then we could proceed to find the integral like we did in the examples above, by replacing 2x\ dx with du and the square root part with sqrt u.

However, the problem int_0^1sqrt(x^2+1)\ dx does not have a "2x" outside of the square root so I cannot use the "u" substitution.

For now, we need to use numerical approaches to evaluate the integral

int_0^1sqrt(x^2+1)\ dx

(Note: Historically, all definite integrals were approximated using numerical methods before Newton and Leibniz developed the integration methods we have learned so far in this chapter.)

We can use two different numerical methods for evaluating an integral:

We meet these methods in the next 2 sections.

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