Section 7.7 Approximate Integration

Hello everyone.
This section is called Approximate Integration. It covers three methods of approximating integrals, but only two of them are covered on the AP Exam.

The first method is the Midpoint Rule. We have covered this before, but just in case you cannot remember it, here it is:

where (for all of the equations).

This calculates the area under a curve using rectangle approximations, using the midpoints as the heights. A more accurate method is trapezoidal approximation. Instead of drawing rectangles under the curve, you can draw trapezoids. Here is the formula for using trapezoidal approximation.

Finally, there is a third method that is not on the AP Exam, and it is known as the Simpson’s Rule. Here it is:

For this to work, n must be even.

Here is an example:

For use the Trapezoidal Rule, the Midpoint Rule, and Simpson’s Rule to approximate the integral if n = 4.

For the Trapezoidal Rule, plug in the values for the equation. First calculate .

Now:

This method gives us an approximation of 44.

Using the midpoint rule, will be the same. Next find the midpoints.

Now plug into the equation:

This gives us an approximation of 21.Finally, for Simpson’s Rule. remains the same.

Plug into the equation:

This gives us an approximation of .

And that's the lesson.

Here is a link to a helpful resource: This website shows the Trapezoidal Approximation and Simpson’s Rule and how to use them: http://archives.math.utk.edu/visual.calculus/4/approx.1/index.html

Sonia you are next.

On a lighter note, here are some comics: