(a) Estimate the area under the graph of f(x)=sinx from x=0...

(a) Estimate the area under the graph of $f(x)=\sin x$ from $x=0$ to $x=\pi / 2$ using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate?
(b) Repeat part (a) using left endpoints.

Key Concepts

Riemann Sums

A Riemann sum is a method for approximating the total area under a curve by summing the areas of multiple rectangles, each of which approximates the area over a small subinterval of the given range. This concept is foundational in understanding and calculating definite integrals, and serves as an introduction to the formal definition of integration.

Partitioning an Interval

Partitioning an interval involves dividing the interval of integration into smaller subintervals of equal or varying lengths. Each subinterval is used to construct a rectangle whose height is defined by the function's value at a specific point, which is crucial for the calculation of a Riemann sum.

Left Endpoint Riemann Sum

A left endpoint Riemann sum uses the value of the function at the left end of each subinterval as the height of the rectangle over that subinterval. This approach is one of several sampling methods used to approximate the area under a curve, and its accuracy depends on the behavior of the function over the interval.

Right Endpoint Riemann Sum

A right endpoint Riemann sum selects the function's value at the right end of each subinterval to determine the height of the corresponding rectangle. Like the left endpoint method, it approximates the area under the curve and its performance (overestimate or underestimate) depends on whether the function is increasing or decreasing.

Underestimate and Overestimate in Riemann Sums

When using Riemann sums, the approximation may either underestimate or overestimate the actual definite integral. This depends on the function's monotonicity: if a function is increasing, a left endpoint sum typically underestimates while a right endpoint sum overestimates the area, and vice versa for a decreasing function. Understanding this concept is key to analyzing the error in numerical integration methods.

Transcript

00:01 This problem serves as a great precursor to integration because this really gets at the heart of the definition of an integral, which is finding the area under the curve.

00:12 And you might have heard of the term or rhyme in sum.

00:17 That's what this problem is centering around.

00:20 Essentially what we do is we take the area under a curve, which we're given the function.

00:25 We split it up into a certain number of triangles.

00:28 We find the area of those triangles and we add them all together and we get the area under the curve.

00:35 So the first thing that we need to do is graph the function we were given.

00:39 We were given a trigonometric function and this graph shows what our function would look like.

00:45 And i've separated it into the four triangles that we need.

00:50 So the first thing that we need to do is estimate the area of these blue rectangles under the curve using right endpoints.

00:59 So the way that we're going to do that would look like this.

01:02 We'd have our area a would be equal to pi over 8 times a sign of pi over 8, plus pi over 8 times a sign of 2 pi over 8, plus pi over 8 times a sign of 4 pi over 8.

01:22 And then you can plug that into a calculator and you would find that the area is 1 .183.

01:28 Now is this an overestimate or an underestimate? well, you can look at the graph, and you can see that our points are not near 1 .1...

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