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Chapter 5: Problem 4

Explain how Riemann sum approximations to the area of a region under a curve change as the number of subintervals increases.

Short Answer

Expert verified
As the number of subintervals increases, the Riemann sum approximation of the area under a curve becomes more accurate. This is because a larger number of smaller width rectangles can better capture the curve's shape, leading to a more accurate estimation of the enclosed area.

Step by step solution

01

Understanding the problem

In order to approximate the area of a region under a curve, we use the Riemann sum. The Riemann sum is a method for approximating the definite integral of a function over a specified interval using a series of subintervals (rectangles) with heights determined by the function's values. The more subintervals we use (smaller width), the more accurate our approximation will be.
02

Example function

Let's consider an example function to illustrate our discussion: f(x) = x^2 on the interval [0, 2]. Our task is to approximate the area under this curve as the number of subintervals increases.
03

Using left endpoints

To follow an orderly process, let's choose the left rule approach for the Riemann sum, which means using the left side of each subinterval as endpoints for our rectangles. We will start with dividing our interval into n subintervals, where n will increase to show how the approximation changes.
04

Defining subintervals and width

The width of each subinterval will be calculated as follows: 螖x = (b-a)/n, where a and b are the limits of our interval [a, b]. In our example, a = 0 and b = 2, so 螖x = (2-0)/n = 2/n. Now, we can move on to calculating the Riemann sum for different values of n.
05

Calculating Riemann sum for different values of n

For n=1, we use only one rectangle, with the height determined by the left endpoint (x=0). In this case, L(1) = f(0) * 螖x = 0 * (2/1) = 0. For n=2, we have two rectangles. First rectangle has height f(x)=0, while the second one has height f(1)=1. By adding their areas, we obtain: L(2) = [f(0) + f(1)] * 螖x = [0 + 1] * (2/2) = 1. For n=4, our rectangles will have a total area of: L(4) = [f(0) + f(1/2) + f(1) + f(3/2)] * 螖x = [0 + 1/4 + 1 + 9/4] * (1/2) = 1.375. As we can see, as the number of subintervals increases, our approximation becomes more accurate.
06

Conclusion

In conclusion, the Riemann sum approximation of the area of a region under a curve becomes more accurate as the number of subintervals increases. This is because a larger number of smaller width rectangles can better capture the curve's shape, leading to a more accurate estimation of the enclosed area.

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