(a) Estimate the area under the graph of f(x)=25-x^2 from...

(a) Estimate the area under the graph of $f(x)=25-x^2$ from $x=0$ to $x=5$ using five 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

Riemann sums are a numerical integration method used to approximate the area under a curve by summing the areas of a collection of rectangles. The heights of these rectangles are determined by the value of the function at specific points within each subinterval, and the accuracy of the approximation depends on the number of subintervals (rectangles) used.

Partitioning the Interval

Partitioning the interval means dividing the total interval over which the area is to be estimated into smaller, equal-width subintervals. This step is crucial as it sets the basis for constructing the individual rectangles whose areas are summed to approximate the total area under the curve.

Right Endpoint Approximation

The right endpoint approximation is a technique where the height of each approximating rectangle is determined by the function's value at the right end of each subinterval. This method is particularly useful when analyzing the behavior of the function, as it can result in either an overestimate or underestimate depending on the slope and curvature of the function.

Left Endpoint Approximation

Left endpoint approximation involves taking the function's value at the left end of each subinterval to determine the height of the corresponding rectangle. Similar to the right endpoint method, this approach can lead to an overestimate or underestimate of the area under the curve, influenced by the function's increase or decrease over the interval.

Underestimate and Overestimate in Numerical Integration

The choice between using right or left endpoints can result in an underestimate or overestimate of the true area under the curve. This outcome depends on the function's monotonicity or concavity. For instance, if a function is decreasing over the interval, using right endpoints may yield an underestimate, while using left endpoints might result in an overestimate. Understanding this behavior is key to analyzing the error in numerical integration methods.

Transcript

00:01 Welcome back to numerad.

00:01 My name is kevin chirac.

00:02 Let's see if we can estimate the area underneath the graph that is f of x equals 25 minus x squared.

00:08 So 25 minus x squared is our function.

00:14 We're going to be looking at between x equals 0 and x equals 5.

00:18 So if i'm just going to sketch this out, this is going to be a downwards facing parabola that comes down and contacts the x -axis at 5 and negative 5.

00:26 And i'm just going to be looking for between 0 and 5.

00:29 So i'm going to be looking at this space over here.

00:30 One, two, three, four, and five.

00:32 Let's start.

00:33 Off by doing right endpoint rectangles.

00:37 So for example, this first one is going to be where i come up to the function and then i cut across.

00:42 I'm plugging in a value of one to get that height.

00:44 So this is going to be five rectangles.

00:47 So we'll do r sub five and say that's going to be equal to what happens when i get a height of by plugging in one times the width.

00:54 And the width here is going to just simply be one.

00:55 And then i'm going to add these up all the way through until i get to f of, well, my last one here is going to be at five, right? i put 5 times 1.

01:06 All right, let's start to add these up.

01:08 If i put in 1 and i square it, i get 24.

01:11 25 minus 1 is 24.

01:13 So it'll be 24, and i'm just going to drop these times 1.

01:15 So they turn out not to matter in this problem here...

Please give Ace some feedback