Approximating Antiderivatives

Can we integrate all continuous functions? Most of the functions that we have been dealing with are what are called elementary functions. These are the polynomials, rational functions, exponential functions, logarithmic functions, trigonometric and inverse trigonometric functions, and all functions that can be obtained from these by the five operations of addition, subtraction, multiplication, division, and composition. If f is an elementary function, then f ’ is an elementary function, but its antiderivative need not be an elementary function. Example, In fact, the majority of elementary functions don’t have elementary antiderivatives. How to find definite integrals for those functions? Approximate!

Approximating definite integrals: Riemann Sums Taking more division points or subintervals in the Riemann sums, the approximation of the area of the domain under the graph of f becomes better. Recall that the definite integral is defined as a limit of Riemann sums. A Riemann sum for the integral of a function f over the interval [a,b] is obtained by first dividing the interval [a,b] into subintervals and then placing a rectangle, as shown below, over each subinterval. The corresponding Riemann sum is the combined area of the green rectangles. The height of the rectangle over some given subinterval is the value of the function f at some point of the subinterval. This point can be chosen freely.

Approximating definite integrals: different choices for the sample points Recall that where x i * is any point in the ith subinterval [x i-1,x i ]. If x i * is chosen to be the left endpoint of the interval, then x i * = x i-1 and we have If x i * is chosen to be the right endpoint of the interval, then x i * = x i and we have L n and R n are called the left endpoint approximation and right endpoint approximation, respectively.

Actual area under curve: First find the exact value using definite integrals. Example

Left endpoint approximation: Approximate area: (too low)

Approximate area: Right endpoint approximation: (too high) Averaging the right and left endpoint approximations: (closer to the actual value)

Averaging the areas of the two rectangles is the same as taking the area of the trapezoid above the subinterval.

Trapezoidal rule This gives us a better approximation than either left or right rectangles.

Can also apply midpoint approximation: choose the midpoint of the subinterval as the sample point. Approximate area: The midpoint rule gives a closer approximation than the trapezoidal rule, but in the opposite direction.

Midpoint rule

Midpoint Rule: (lower than the exact value ) 0.625% error Trapezoidal Rule: 1.25% error (higher than the exact value) Notice that the trapezoidal rule gives us an answer that has twice as much error as the midpoint rule, but in the opposite direction. If we use a weighted average: This is the exact answer!

This weighted approximation gives us a closer approximation than the midpoint or trapezoidal rules. Midpoint: Trapezoidal: twice midpointtrapezoidal

Simpson’s rule Simpson’s rule can also be interpreted as fitting parabolas to sections of the curve. Simpson’s rule will usually give a very good approximation with relatively few subintervals.

Example:

Error bounds for the approximation methods Examples of error estimations on the board.