Approximate Integration Techniques

Definite Integral

The limit of a Riemann sum as the width of the intervals approaches zero

Evaluation Theorem

Requires knowing an antiderivative to evaluate a definite integral

Antiderivative

A function whose derivative is the integrand

Numerical Integration

Techniques used to find approximate values of definite integrals

Midpoint Rule

Estimates a definite integral using midpoints of subintervals

Trapezoidal Rule

Approximates a definite integral using trapezoids

Error Bounds

Amount added to an approximation to make it exact

Simpson's Rule

Uses parabolas to approximate a curve for integration

Approximation Error

The difference between an approximation and the exact value of an integral

Riemann Sums

An approximation method for finding the area under a curve

Midpoint Approximation

Estimates a definite integral using midpoints of subintervals

Trapezoidal Approximation

Estimates a definite integral using trapezoids instead of rectangles

Simpson's Approximation

Estimates a definite integral using parabolas instead of straight line segments

Error Bound Theorem

Provides upper bounds for errors in approximating definite integrals

Midpoint Error Bound

Upper bound for error in using the midpoint rule for integration

Trapezoidal Error Bound

Upper bound for error in using the trapezoidal rule for integration

Simpson's Error Bound

Upper bound for error in using Simpson's rule for integration

Number of Intervals

Determines the accuracy of an estimate for a definite integral

Homework Assignment

Problems to practice concepts from Section 5.9 on approximate integration