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The Integral And The Area Under A Curve. How do we find the area between a curve and the x- axis from x = a to x = b ?. We could approximate it with rectangles:. We could use more than one rectangle to approximate the area.

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The Integral

And

The Area Under A Curve





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Area under the curve is approximately equal to the sum of the areas of the rectangles.

More rectangles = better approximation of Area


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Here’s a demonstration of how the approximation works: the areas of the rectangles.

http://www.slu.edu/classes/maymk/Riemann/Riemann.html


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Riemann Sums the areas of the rectangles.

Riemann Sums help us to make the calculation of the area under a curve more uniform (or easier):


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Riemann Sums the areas of the rectangles.

  • First divide the interval into equal parts

  • Then we choose a point in each interval to make a rectangle

  • Use the chose x* value to find the height of each rectangle by simply finding f(x*) for each interval.


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Riemann Sums the areas of the rectangles.

We can use the right endpoints of each subinterval as the x*


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Riemann Sums the areas of the rectangles.

Or we can use the left endpoints of each subinterval as the x*


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Riemann Sums the areas of the rectangles.

Or we can use the midpoints of each subinterval as the x*


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Riemann Sums the areas of the rectangles.

Using left endpoints, our calculations, we proceed as follows:

Width of each rectangle:

In this case:

In general:


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Riemann Sums the areas of the rectangles.

Using left endpoints, our calculations proceed as follows:

Height of each rectangle =

For example, the height of the third rectangle is:

Area of each rectangle =


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Riemann Sums the areas of the rectangles.

Using right endpoints, our calculations, we proceed as follows:

Width of each rectangle:

In this case:

In general:


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Riemann Sums the areas of the rectangles.

Using right endpoints, our calculations proceed as follows:

Height of each rectangle =

For example, the height of the fourth rectangle is:

Area of each rectangle =


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Riemann Sums the areas of the rectangles.

Using midpoints, our calculations, we proceed as follows:

Width of each rectangle:

In this case:

In general:


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Riemann Sums the areas of the rectangles.

Using midpoints, our calculations proceed as follows:

Height of each rectangle =

For example, the height of the second rectangle is:

Area of each rectangle =


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The greater the value of the areas of the rectangles.n used, then the better the approximation of the area will be:

We say:

Note: Depends on whether we use left endpoints, right endpoints, or midpoints.


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Here’s another demonstration of how the approximation works:

http://www.slu.edu/classes/maymk/Riemann/Riemann.html


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