6.3 Numeric Integration

Motivate WHY - Gives an approximate value for a definite integral when can’t or difficult to find “antiderivative”

#4

Use Riemann sums with Rectangles using Right hand endpoints, left hand endpoints, and midpoints.

R₄ Right Hand Sum for n = 4

L₄ Left Hand Sum for n = 4

A₄ Average of Left and Right Sum

M₄ Midpoint Sum for n = 4

Online Links to visual Riemann Sums

http://www.slu.edu/classes/maymk/Applets/Riemann.html

http://calculusapplets.com/riemann.html

http://math.scu.edu/~dsmolars/ma12/riemann.html

Trapezoidal Rule Recall Area formula for trapezoid

For one Trapezoid this becomes

So if n = 6, Add up 6 trapezoids,

Factor out

Generalize to get Trapezoidal Rule:

Use trapezoidal rule for same problem as above with rectangular approximations.

#4

Error Estimate for Trapezoidal Rule where M is the maximum value of on [a,b}

Find the Maximum error for #24 using the Trapezoidal Rule

Simpson’s Rule Approximates area using parabola – n should be an even number

Use Simpson’s Rule to find:

Error Estimate for Simpson’s Rule where M is the maximum value of on [a,b]

Find Error estimate on #24 using Simpson’s Rule

Approximate with Trapezoid, Simpson’s, and find error estimates

#18

Find value of n for required accuracy

Determine how many subintervals are required to guarantee accuracy to within 0.00005 in the approximation of the integral by a) the trapezoidal rule and b) Simpson’s Rule.

#22

Assignment: Pg. 507; 3-11 odd, 17-23 odd

[Calculator 29, 31, 37, 41]