1. Trapezoidal Approximations Greg Kelly, Hanford High School, Richland, Washington Modified: Mike Efram Healdsburg High School 2004 Photo by Vickie Kelly, 2002 Greenfield Village, Michigan

2. Using integrals to find area works extremely well as long as we can find the antiderivative of the function. Sometimes, the function is too complicated to find the antiderivative. At other times, we don’t even have a function, but only measurements taken from real life. What we need is an efficient method to estimate area when we can not find the antiderivative.

3. Left-hand rectangular approximation: Approximate area: (too low)

4. Approximate area: Right-hand rectangular approximation: (too high)

5. Averaging right & left values of the function gives trapezoids: Trapezoidal approximation

6. (still too high… but much better)

7. Ex.