In Exercises 15-18, use a finite sum to estimate the average...

In Exercises 15-18, use a finite sum to estimate the average value of $f$
on the given interval by partitioning the interval into four subintervals
of equal length and evaluating $f$ at the subinterval midpoints.
$$
f(t)=(1 / 2)+\sin ^{2} \pi t \quad \text { on } \quad[0,2]
$$

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Key Concepts

Equal Partitioning of an Interval

Dividing an interval into equal subintervals is a foundational step in many numerical methods, including the midpoint rule. This systematic partitioning simplifies the calculation by ensuring each subinterval has the same width, which is then used to weight the corresponding function evaluations uniformly.

Average Value of a Function

This concept involves determining the mean value of a function over a specified interval, typically by computing the integral of the function over that interval and then dividing by the interval's length. It provides an overall measure of the function's behavior across the range.

Numerical Integration

Numerical integration is a method for approximating the definite integral of a function when an analytic solution is challenging or impossible to obtain. It relies on summing weighted function values at specific points within the interval to estimate the area under the curve.

Midpoint Rule

The midpoint rule is a specific numerical integration technique where the function is evaluated at the midpoint of each subinterval. The average of these values, multiplied by the width of the subintervals, approximates the integral over the entire interval, offering a balance between simplicity and accuracy.

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