Applications of Integration

1. Applications of Integration • Hanna Kim & Agatha Wuh • Block A

2. Disk Method Intro Disk Method is in many ways similar to cutting a rollcake into infinitely thin pieces and adding all of them together to find out the volume of the whole.

3. Delving into Disk Method • Disk = right cylinder • Volume of disk= (area of disk) * (width of disk)=πr2w • Integration is simply an addition of infinitesimally thin disks in a given interval. Demonstration of how the whole volume is computed by the addition of smaller ones

4. Disk Method Integration Horizontal Axis of Revolution Vertical Axis of Revolution πr2=Area of Circle

5. Disk Method Example The area under the curve y is rotated about the x-axis in the interval [-π/2, 3π/2]. Find its volume.

6. Washer Method Intro Washer’s method is an extension of disk method. However, the difference is that there is a hole in the middle just like a bamboo or a collection of CDs!

7. Delving into Washer Method • Washer=a disk with a hole • Using this method is to find out an area of washer by subtracting the smaller circle from the bigger circle and then adding all the washers up.

8. Washer Method Integration

9. Washer Method Example The region bounded by y=-5(x-3)2+8 and y=sin x +2 is revolved around the x-axis. Find its volume. Find points of intersection 1. Graph the two equations. 2. Press 2nd Trace or Calc 3. Press 5: intersect 4. Identify the two different graphs. 5. Done!

10. Washer Method Example Outer circle Inner circle

11. Shell Method Intro Shell Method is like “unrolling” a toilet paper and adding up the infinitely thin layers of paper.

12. Delving into Shell Method Volume of shell= volume of cylinder-volume of hole=2πrhw 2π(average radius)(height)(thickness) 2πrh=circumference*height =Area of cylinder without the top and bottom circles of the cylinder

13. Shell Method Integration

14. Shell Method Example The region bounded by the curve , the x-axis, and the line x=9 is revolved about the x-axis to generate a solid. Find the volume of the solid. (9,3) The integral is in terms of y because y is parallel to the axis of rotation. radius height

15. Shell Method Example Constant multiple rule Evaluate from 0 to 3 units3 Answer

16. Section Method Intro The section method is like cutting a loaf of bread into different slices and adding those slices up.

17. Delving into Section Method • Utilizing simple area equation of the cross section. • Formula is applicable to any cross section shape • Most common cross sections are squares, rectangles, triangles, semicircles, trapezoids

18. Section Method Integration Cross section taken perpendicular to the x-axis Cross section taken perpendicular to the y-axis

19. Section Method Example The solid lies between planes perpendicular to the y-axis at y=0 and y=2. The cross sections perpendicular to the y-axis are half circles with diameters ranging from the y-axis to the parabola The integral is in terms of y because the cross sections are perpendicular to the y-axis.

20. Section Method Example Evaluate 1/2 πr2=half circle Simplify Constant multiple rule units3 Answer

21. Shell Method VS. Disk Method

22. How Can We Distinguish? • For disk method, the rectangle (disk) is always perpendicular to the axis of revolution • For shell method, the rectangle is always parallel to the axis of revolution.

23. Works Cited • http://library.thinkquest.org/3616/Calc/S3/TSM.html • http://mathdemos.gcsu.edu/mathdemos/shellmethod/2curvesshells.gif • http://mathdemos.gcsu.edu/mathdemos/sectionmethod/sectionmethod.html • http://mathdemos.gcsu.edu/mathdemos/washermethod/