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.
$$f(t)=1-\left(\cos \frac{\pi t}{4}\right)^{4} \text { on } [0,4]$$

Key Concepts

Midpoint Rule

The midpoint rule is a specific type of Riemann sum where the function is evaluated at the midpoint of each subinterval, rather than at the endpoints. This approach often yields a better approximation of the integral because midpoints can more accurately capture the behavior of the function within each subinterval.

Average Value of a Function

This concept involves determining the mean value of a function over a specified interval, which is typically calculated by dividing the integral of the function over the interval by the length of the interval. It provides a measure of the 'typical' value assumed by the function on that interval.

Riemann Sum

A Riemann sum is a method for approximating the value of a definite integral by summing the areas of a collection of rectangles that approximate the area under a curve. The accuracy of this approximation depends on the size and number of these subintervals.

Partitioning the Interval

Partitioning an interval refers to dividing the interval into smaller, equally spaced subintervals. This process is essential in numerical integration methods like Riemann sums, as it defines the sampling points and determines the granularity of the approximation.

Transcript

00:01 So this question is asking us to find the average value of f on the given interval by partitioning the interval into four subintervals of equal length and evaluating f at the sub interval points.

00:11 So first, the question is asking us to find the average value of f, and we know from the book that that is approximately the area divided by the length of our interval.

00:32 So our first step here is we do need to find the area.

00:38 And for this one it is telling us to find it by the midpoint.

00:42 So we're wanting to separate this into four different sub intervals.

00:47 So i see my whole intervals from zero to four.

00:50 So that means my delta x is just going to equal one.

00:57 So if i separate that into four sub intervals, going by one each time, i'm just going to have zero to one, one to two, three to four, not three to four, two to three, and then three to four.

01:25 So first i want to find my midpoints in there.

01:29 So halfway between zero and one, you want half, halfway between one and two is going to be three halves, halfway between two and three and three is going to be five halves, and then halfway between three and four is going to be seven halves.

01:50 So i have my width of my rectangles.

01:54 I know where my heights are going to happen at.

01:57 So now i just need to plug in my ms into the function to actually find my height that x value...

Please give Ace some feedback