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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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slide1

The Integral

And

The Area Under A Curve

slide5

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

More rectangles = better approximation of Area

slide6

Here’s a demonstration of how the approximation works:

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

slide7

Riemann Sums

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

slide8

Riemann Sums

  • 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.
slide9

Riemann Sums

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

slide10

Riemann Sums

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

slide11

Riemann Sums

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

slide12

Riemann Sums

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

Width of each rectangle:

In this case:

In general:

slide13

Riemann Sums

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 =

slide14

Riemann Sums

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

Width of each rectangle:

In this case:

In general:

slide15

Riemann Sums

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 =

slide16

Riemann Sums

Using midpoints, our calculations, we proceed as follows:

Width of each rectangle:

In this case:

In general:

slide17

Riemann Sums

Using midpoints, our calculations proceed as follows:

Height of each rectangle =

For example, the height of the second rectangle is:

Area of each rectangle =

slide18

The greater the value of 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.

slide19

Here’s another demonstration of how the approximation works:

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