Chapter Contents

• Integration
• 1. The Differential
• 2. Antiderivatives and The Indefinite Integral
• 3. The Area Under a Curve
• 4. The Definite Integral
• 5. Trapezoidal Rule
• 6. Simpson’s Rule
• Riemann Sums
• Integration Mini-lectures
• The Differential
• Difference Between Differentiation and Integration
• Given dy/dx, find y = f(x)
• Integration by Substitution
• Difference Between Definite and Indefinite Integrals
• Area Under a Curve
• Millionaire Calculus Game
• Integration Problem Solver

• Comments, Questions?

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From the math blog...

2. Antiderivatives and The Indefinite Integral

by M. Bourne

We wish to perform the opposite process to differentiation. This is called "antidifferentiation" and later, we will call it "integration".

Example

If we know that

and we need to know the function this derivative came from, then we "undo" the differentiation process. (Think: "What would I have to differentiate to get this result?")

y = x³ is ONE antiderivative of

There are infinitely many other antiderivatives which would also work:

• y = x³ + 4
• y = x³ + π
• y = x³ + 27.3

In general: y = x³ + K, is the indefinite integral. K is called the constant of integration.

In electrical engineering, we prefer to write "+K", since C (as used in most math texts) is used for capacitance.

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We write: and say:

"The integral of 3x² with respect to x equals x³ + K."

Here is a template you can play with to get the idea.

LIVEMath

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The sign is an elongated "S", standing for "sum". Later we will see that the integral is the sum of the areas of infinitely thin rectangles.

Loading Flash movie...

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Note: Sometimes we write: F(x) to mean the integral of f(x). So we have:

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Exercise: Find

Answer

Note:In general, we can use:

1. (k,K are constants.)

The integral of a constant is that constant times x.

Example:

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2. (n ≠ -1)

For the integral of a power of x: add 1 to the power and divide by the new number.

Example:

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DON'T FORGET THE "+ K" (or "+ C"). THIS CONSTANT OF INTEGRATION IS VITAL IN LATER APPLICATIONS.

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Let's see 2 examples in Flash:

1. How to do basic integration:

Loading Flash movie...

2. Another example:

Loading Flash movie...

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Example 1:

Answer

Example 2:

Answer

Example 3:

Answer

Example 4: A particular curve has .

We are told that the curve passes through the point (2, 5). Find the equation of the curve.

Answer

We can see an animation of the resulting family of curves in this LiveMath document:

LIVEMath

Example 5:

This is different to the other exercises above! It is a function of a function situation, so we have to do the reverse of the Chain Rule, which we met in the section on differentiation.

The new rule we require is...

Power Formula for Integration

(n ≠ -1)

This requires a substitution step, where u(x) is some function of x.

Now back to the problem to see how to apply this formula:

Answer

Let's see how it works in Flash:

Loading Flash movie...

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Example 6:

Answer

Example 7: Find using a substitution.

Answer

Example 8: Given , find the function y = f(x) which passes through the point (0,2).

Answer

Note: You will see "+K" and "+C" in this work. Most textbooks use + C.

Always use +K if you are answering electrical problems.