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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1. The Differential

Earlier in the differentiation chapter, we wrote and f '(x) to mean the same thing. We used as an operator.

We now see a different way to write, and to think about, the derivative.

Definition:

The differential of y = f(x) is written:

dy = f '(x)dx.

Note: We are now treating as a fraction.

Example:

Find the differential of y = 3x⁵- x.

 

Answer

dy = f'(x)dx

dy = (15x⁴ - 1)dx

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Note: We could use the differential to estimate the real change in value of a function (Δy) caused by a small change in x (written as Δx). Many text books do this, but it is pretty silly, since we can easily find the exact change - why approximate it?

We are introducing differentials here as an introduction to the notation used in integration.

 

How are dy, dx and Δy and Δx related?

See Slope of a tangent for some background on this.

Relative Error

Note: We may perform a calculation and find an error of 1 cm. If this is part of a problem involving lengths of 8 cm, then we have a big problem. If it involves lengths of 3 km, we have an insignificant problem. We usually talk about the relative error:

or

rather than the absolute error of 1 cm.

 

Proportionate and Percentage Change

The proportionate change in y(x) is given by (no units).

The percentage change in y(x) is given by