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Without using a spreadsheet, determine the approximate area of the region bounded by the given curve, the $x$ -axis and between the given lines, using the indicated number $n$ of rectangles.$f(x)=x^{3},$ between $x=2$ and $x=3, n=4$ using the (a) left endpoint, (b) right endpoint, (c) midpoint.

(a) 13.953125(b) 18.703125(c) 16.210938

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 4

Approximation of Areas

Integrals

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

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Approximate the area under…

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'Approximate the area…

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So we're working here with the function F of X is equal to three plus x and we're gonna be looking for Is the area under this curve over an interval? What's the interval we're looking at? Well, in this case, we're looking at the interval from 1 to 3. One, 23 Well, 123 Now, if I'm looking at the interval from 1 to 3 and we know given the directions that I have to have four rectangles to calculate the area. When I split the distance between 1 to 3 evenly, it will make each interval half a unit half one half to so we know the width of the rectangles will be a half now in order work with Although the left side the right side, the middle What I'm gonna do is I'm gonna create a table of values off all of the output values at each of the interval points at one at 1.5 at two at 2.5 and a three substituting one into the equation. I'm going to get one plus three is four. 1.5 plus three is 4.5. And following that pattern, we're going to have five, 5.5 and six. I'm gonna use these output values as I work through my problem. The first thing we're going to do is focus on the left hand side, used using the rectangles based on the left hand side while my rectangles will occur on each of the intervals. But with of the rectangles will be a half. And what I'm gonna uses the height at the left hand side of each of the rectangles, which will be the output value at one 1.5 2 And it's you and a half, which will be four plus 4.5 plus five plus 5.5. So for plus 4.5 plus five, those 5.5 is 19. So you're gonna go with 19 halves, which is the same as 9.5 now, using the right hand side approximation. But with the rectangles does not change. It's still going to be a half, except now I'm gonna focus on the output values on the right side of the rectangle at 1.5 to 2.5 and a three, which means it will be 4.5 plus five plus 5.5 plus six, adding those together I'm going to get 21 divided by 2 21 has, which is 10.5. Averaging those two values nine and halfway between nine half and 10.5 is going to be so. Finally, we want to calculate the midpoint method. My distance of my rectangle is still going to be a half. The base will still be a half, but I'm now going to use the values in the middle of. Of each of the intervals well, halfway between four and 4.5 is for in a quarter. Halfway between 4.5 and five is four and three quarters, five and a quarter in five and three quarters. Adding those together 4.25 plus 4.75 plus 5.25 plus 5.75 will give us 20 divided by two, which is 10. Notice that the average in my midpoint value R equals

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