In the following exercises, estimate the areas under the curves by computing the left Riemann su

In the following exercises, estimate the areas under the...

Key Concepts

Riemann Sum

A Riemann sum is a method for approximating the value of a definite integral by summing the areas of simple shapes (usually rectangles) that approximate the area under a curve. This method relies on partitioning the domain of the function into small subintervals and summing the products of the function values at chosen points and the subinterval widths.

Left Riemann Sum

A left Riemann sum is a specific type of Riemann sum where the function is evaluated at the left endpoint of each subinterval to determine the height of each approximating rectangle. This approach provides an estimation of the area under the curve, and its accuracy depends on the function's behavior and the number of subintervals used.

Partitioning the Interval

Partitioning the interval involves dividing the domain of the function, which is being integrated, into a fixed number of equal or unequal subintervals. This division is crucial for numerical integration methods such as Riemann sums, as it determines the width of each rectangle used in the approximation process.

Transcript

00:01 Okay, for problem number 45, we're going to find the left -hand riemann sum of this given picture.

00:09 And so we're going to divide this picture up into eight rectangles, because that's what the little eight means here.

00:19 And for each of these rectangles, we're going to go up the left side of the rectangle from the x -axis until it hits the curve.

00:30 So right like this.

00:33 Until it hits the curve, and then we're going to go straight over to establish our height.

00:39 And since we need eight rectangles, and we're going from zero to eight, it means that every rectangle is going to have a width of one right here.

00:51 So i'm just going to go over one unit and draw my rectangle.

00:59 Okay, and then the next rectangle just starts where the first one left off, but we need to go up till it hits the curve and then go straight over for one unit and down.

01:13 This is how you graphically set up a left -hand ream and sum.

01:18 We'll go up the left side of this third rectangle till it hits the curve, straight over, down.

01:27 And then the fourth rectangle, if we trace up this side here until it hits the curve, it's going to be a little bit shorter because it hits the curve sooner.

01:40 But the fifth one has got to go all the way up here to hit the curve.

01:46 Okay, come down.

01:49 The next one's going to hit the curve here...

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