Compute the left and right Riemann sums -L6 and R6 respectively - for f(x)=√(9-(x-3)^2) on [0,6]

Compute the left and right Riemann sums -L6 and R6...

Key Concepts

Right Endpoint Approximation

The right endpoint rule is a Riemann sum technique in which the function’s value at the right endpoint of each subinterval is used to calculate the height of the corresponding rectangle. Like the left endpoint rule, the accuracy of this method depends on whether the function is increasing or decreasing, and it is often compared to the left endpoint rule to analyze the approximation error.

Left Endpoint Approximation

The left endpoint rule is a type of Riemann sum where the height of each rectangle is determined by the value of the function at the left endpoint of each subinterval. This method can lead to either an underestimate or an overestimate of the true integral, depending on the behavior of the function over the interval.

Partitioning the Interval

Partitioning an interval involves dividing the interval of integration into a set number of subintervals. The width of each subinterval, often denoted as ?x, plays a critical role in the accuracy of the Riemann sum, as it determines the size of each rectangle used in the approximation.

Riemann Sum

A Riemann sum is a method for approximating the definite integral of a function by summing up the areas of a finite number of rectangles whose heights are determined by the function's values at specific points. This concept forms the foundational approach to defining and understanding integration in calculus.

Transcript

00:01 Okay, this problem, we're going to compare the left endpoint sum, the left ream and sum to the right ream and sum over six sub -intervals for this function on the interval from zero to six.

00:23 Let's write that up here.

00:26 Zero to...

00:30 Okay.

00:31 I've already got a couple pictures drawn over here to help us understand this.

00:36 The left sum is, first of all, we're going to figure out where exactly we have to divide up these rectangles.

00:46 But it's pretty easy to see if we do the entire interval and we divide it up six ways that each sub -interval is going to have a width of one.

01:01 So you can see that here.

01:06 That's why i have the x -axis stepped off this way.

01:09 Because if we just add one to each x value, we'll get the next x value.

01:16 And it's the same way with the right end point sum also.

01:22 So to execute the left ream and sum for six sub -intervals, we just want to find the area of all six of these rectangles.

01:37 It's like we have a rectangle here, here.

01:43 And we actually have one here that you can't really see because the left hand side of it has a function value, a y value of zero.

01:56 So if we find an area of a rectangle with a y value of zero, a height of zero, then that area is just going to be zero.

02:06 So that's going to be our first rectangle on the left hand riemann sum.

02:11 They all have a width of one.

02:15 Actually, we'll go ahead and use the distributive property there.

02:19 A width of one.

02:21 And then we just need to find the height of each of these rectangles.

02:28 So we know our first height's going to be zero.

02:33 Our next height is going to be whatever the function value is at one.

02:42 Okay, at this x value.

02:45 If we plug that x value into the function, then we'll get this value right here, which is the height of this rectangle on the left -hand side.

02:57 So if we plug a 1 in here, we're going to get 1 minus 3 is negative 2, and negative 2 squared is going to be 4, and 9 minus 4 is going to be 5.

03:11 So that's going to be root 5.

03:18 Okay.

03:19 Now the height of my third rectangle, this rectangle here, is going to be the function value at 2 will give me this point.

03:29 So if we evaluate this function at 2, 2 minus 3, is negative 1.

03:35 Negative 1 squared is 1, 9 minus 1 is 8.

03:38 So this is root 8.

03:43 And we're going to continue right on down the line.

03:47 We're going to get this next function value at 3 for the fourth rectangle.

03:58 So that one is going to be 3 minus 0, 9 minus 0 is 9 and the root 9 is going to be 3.

04:09 Now we suspect this picture has some symmetry about x equals 3...

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