1 / 20

The Integral And The Area Under A Curve

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.

desma
Download Presentation

The Integral And The Area Under A Curve

An Image/Link below is provided (as is) to download presentation 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. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. The Integral And The Area Under A Curve

  2. How do we find the area between a curve and the x-axis from x = a to x = b?

  3. We could approximate it with rectangles:

  4. We could use more than one rectangle to approximate the area

  5. Area under the curve is approximately equal to the sum of the areas of the rectangles. More rectangles = better approximation of Area

  6. Here’s a demonstration of how the approximation works: http://www.slu.edu/classes/maymk/Riemann/Riemann.html

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

  8. 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.

  9. Riemann Sums We can use the right endpoints of each subinterval as the x*

  10. Riemann Sums Or we can use the left endpoints of each subinterval as the x*

  11. Riemann Sums Or we can use the midpoints of each subinterval as the x*

  12. Riemann Sums Using left endpoints, our calculations, we proceed as follows: Width of each rectangle: In this case: In general:

  13. 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 =

  14. Riemann Sums Using right endpoints, our calculations, we proceed as follows: Width of each rectangle: In this case: In general:

  15. 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 =

  16. Riemann Sums Using midpoints, our calculations, we proceed as follows: Width of each rectangle: In this case: In general:

  17. 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 =

  18. 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.

  19. Here’s another demonstration of how the approximation works: http://www.slu.edu/classes/maymk/Riemann/Riemann.html

More Related