[T] Use a computer algebra system to compute the Riemann sum, LN, for N=10,30,50 for f(x)=(1

[T] Use a computer algebra system to compute the Riemann...

Key Concepts

Numerical Integration

Numerical integration refers to the process of calculating the value of a definite integral using numerical techniques, such as Riemann sums, rather than finding an exact antiderivative. This is particularly useful when the function does not have an elementary antiderivative, or when an approximation is sufficient for practical purposes.

Left-Endpoint Approximation

The left-endpoint approximation is a specific form of the Riemann sum where the function value used to calculate the height of each rectangle is taken at the left endpoint of each subinterval. This method is simple to implement but may either overestimate or underestimate the true area under the curve depending on the function's behavior.

Computer Algebra System

A computer algebra system (CAS) is software designed to perform symbolic mathematics, including operations like simplifying expressions, solving equations, and calculating integrals both symbolically and numerically. Such systems are valuable tools for handling computations in calculus, especially when dealing with complex functions or when automation of repetitive calculations, such as computing Riemann sums for various partition sizes, is needed.

Riemann Sum

A Riemann sum is a method for approximating the total area under a curve (integral) by summing up areas of simple geometric shapes, typically rectangles, whose heights are determined by the value of the function at selected points within subintervals of the overall interval. This approach is foundational in integral calculus and serves as the basis for defining the definite integral.

Partitioning of an Interval

Partitioning an interval involves dividing the interval into a specified number of subintervals, usually of equal length, to facilitate the approximation of an integral. The choice of partition size affects the accuracy of the Riemann sum, with finer partitions generally leading to a better approximation of the area under the curve.

Transcript

00:01 So for this problem, we're going to be computing a few different riemann sums, and they're going to all be the left -hand sums for n equals 10, 30, and 50 sub -intervals.

00:11 And so we can go ahead and start with our minimum of 10 sub -intervals.

00:15 So we'll go ahead and do this part in red.

00:18 And we know that our first step in calculating a raman sum is going to be finding our delta x or our step size.

00:24 And the way that we do this is we're going to subtract the right end point of our interval.

00:27 So in this case, our interval is from negative 1 to 1.

00:31 So our right end point will be positive 1, and we're going to subtract our left endpoint.

00:37 And then we divide this by whatever our number of sub -intervals is.

00:40 So in this case, our number of sub -intervals, n, is 10.

00:44 So that means that our delta x is going to reduce to 2 over 10 or 1 -5th.

00:50 And from here we're going to be able to calculate what our x sub -i star is.

00:55 And essentially, the way that we figure this out is we just start, at our left endpoint, so in this case negative one.

01:02 We're going to increase it each time by one of our step sizes, which is 1 over 5, times our initial i, because that's going to tell us which step number we're on, minus 1, because of the fact that we have our left endpoints.

01:18 And so from here, we're going to be able to set up our sigma notation.

01:23 So we know that we are going to say that l sub 10 is equal to whatever our delta x is.

01:29 So in this case, one fifth, times sigma going from i is equal to 1, up through our number of sub intervals, which is 10, of f of x of i star...

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