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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Riemann Sums Java Applet

You can use this Java applet to explore the concept of numerical integration. We met this concept before in Trapezoidal Rule and Simpson Rule.

Before integration was developed, the only way to find the area under a curve was to draw rectangles with increasingly smaller widths to get a good approximation.

Remember, we are using the area under a graph to represent some physical quantity.

For example, integration can help us to find a velocity from an acceleration, or to solve problems in electronics.

Things to Do

In this applet, you start with a predefined function that has been drawn for you. You can:

• At the bottom, increase the number of intervals (try 20). Click "Compute" and you get more rectangles and the accuracy of the approximation improves. (The 'correct' answer to 9 decimal places is 8.240404602.)
• Now try different options from the "Method" pull-down menu. Consider which one gives the best approximation and why. You can see Trapezoid Rule in the last option.
• You can change the range of values of x and y.
• You can choose different example functions from the pull-down menu at the top. (Choose the example and then click "Load Example" to the left.)
• Now try your own function in the box at the bottom. Try simple ones like sin(x) or tan(x) first.

Applet by David Eck and team from the Hobart and William Smith Colleges.

Sorry about the slow load time for this applet - it is pretty big...