Definite Integrals - Average Values, Integration Rules?

• starless.aeon
In summary, a definite integral is a mathematical concept that represents the area under a curve between two specific points on a graph. It is denoted by ∫ and is calculated by finding the difference between the upper and lower limits of integration. The average value of a function can be calculated using a definite integral by dividing the integral by the difference between the limits of integration. Some common integration rules include the power rule, substitution rule, integration by parts, and trigonometric substitution. Definite integrals can be evaluated using technology but it is important to understand how to solve them manually. In real-world applications, definite integrals are used to calculate quantities such as displacement, velocity, and area under a curve, as well as finding the average value
starless.aeon

Homework Statement

Find the average value of the function f(t) = (t-3)^2 on [0,6]

Homework Equations

Average value equation (see below)

The Attempt at a Solution

Okay, I have this:

http://img4.imageshack.us/img4/5338/howthecrap.png

But I don't know how it gets to the last step?

Last edited by a moderator:
You did great up until the end. Substitute the values of the three integrals into what you have in the line above. You should get 1/6*(72 - 6*18 + 9*6). This answer, when simplified, agrees with mine.

1. What is a definite integral?

A definite integral is a mathematical concept that represents the area under a curve between two specific points on a graph. It is denoted by ∫, and the two points are called the limits of integration.

2. How is the average value of a function calculated using a definite integral?

The average value of a function can be calculated by dividing the definite integral of the function by the difference between the upper and lower limits of integration.

3. What are some common integration rules?

Some common integration rules include the power rule, substitution rule, integration by parts, and trigonometric substitution. These rules help to simplify the process of solving definite integrals.

4. Can definite integrals be evaluated using technology?

Yes, definite integrals can be evaluated using technology such as calculators and software programs. However, it is important to understand the concepts behind definite integrals and how to solve them manually.

5. How are definite integrals used in real-world applications?

Definite integrals are used in various fields such as physics, engineering, and economics to calculate quantities such as displacement, velocity, and area under a curve. They can also be used to find the average value of a function, which is useful in analyzing data and making predictions.

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