The Integral And The Area Under A Curve

1 / 20

# The Integral And The Area Under A Curve - PowerPoint PPT Presentation

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## The Integral And The Area Under A Curve

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

The Integral

And

The Area Under A Curve

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

More rectangles = better approximation of Area

Here’s a demonstration of how the approximation works:

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

Riemann Sums

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

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.

Riemann Sums

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

Riemann Sums

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

Riemann Sums

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

Riemann Sums

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

Width of each rectangle:

In this case:

In general:

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 =

Riemann Sums

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

Width of each rectangle:

In this case:

In general:

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 =

Riemann Sums

Using midpoints, our calculations, we proceed as follows:

Width of each rectangle:

In this case:

In general:

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 =

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.

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