Well, 2 squared plus 1 isĥ, so it's 5 times 1/2. And then we have theįunction evaluated at 2. To be equal to f of- it's going to be equal to To our approximate area- let me make it clear-Īpproximate area under the curve, just the Orange color- so plus the function evaluatedĪt 2.5 times the base. The third rectangle, the height is the functionĪ different color than what I wanted to use. Third rectangle is going to be the functionĮvaluated at its left boundary, so f of 2- so plus f of 2 We already said was f of 1.5, times the base, times delta x. Second rectangle is going to be the height, which So let's figure out what theĪreas of each of the rectangles are. Many more rectangles, it would be a betterĪpproximation. That in with a color that I have not used. These rectangles? And clearly, this isn't going So the function evaluatedĪt 2.5 is the height. Have our fourth rectangle, the function evaluated at 2.5. Rectangle, we have the function evaluated at 2. We're now looking at theįunction evaluated at 1.5. To the left boundary of the second rectangle. That is the height of our first rectangle. I'm going to use the functionĮvaluated at the left boundary to define the height. Ways of doing this- I'm going to use the leftīoundary of the rectangle to define the height- or We'll see in future videos that there's other Now, let's thinkĪbout how we'll define the height of the rectangles. This first interval between the boundary between theįirst rectangle and the second is going to be 1.5. To be the total distance that we're traveling in x. What those widths have to be if they're goingĬall that delta x. Those rectangles look like, so four rectangles of equal width.

To approximate it by constructing four rectangles Try to do in this video is approximate theĪrea under the curve y is equal to x squared plus 1īetween the intervals x equals 1 and x equals 3.

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