Double & triple integral unit 5 paper 1 , B.Sc. 2 Mathematics

This ppt covers the following topics of unit - 5 of B.SC. 2 Mathematics :- Introduction , Definition & example of Double & Triple Integration.

1. Shri SHANKARACHARYA mahavidyalayA junwani bhilai ADVANCED CALCULUS “ Double and Triple Integral” A.P. PREETI SHRIVASTAVA

2. Introduction Define double integrals Properties of double integrals Example of double integrals Define triple integrals Example of triple integrals index

3. Introduction:- In elementary calculus, students have studied the definite integral of a function of one variable . In this chapeter we show how the notion of the definite integral can be extended to functions of several variables. In particular, we shall discuss the double integral of a functions of two variable and the triple integral of a function of three variables.

4. Define double integrals:- Let f(x,y) be a function of two independent variables x,y in - plane defined at all points in a finite region A. let the region A be divided into n subregions (k=1,2,…,n) of areas let be any point inside the kth element Let us form the sum let us the number of subdivision becomes infinite in such a way that the dimensions of each subdivision approaches to zero .

5. If under these conditions, the limit which is independent of the way in which the point are chosen exists, then this limit is called the double integral of over the region A, written is defined by

6. Properties of double integral :- if f and g are continuous over the bounded region R, then: where R is composed of two subregions R1 and R2 or

7. Example of double integrals:- 1. evaluate: sol:-

8. Example 2- when the region of integration R is the triangle bounded by y= 0, y= x and x = 1 , show that sol . The region of interation is shown shaded in the adjoining figure. Let us divide the triangle OAB into vertical strips. Then it is evident that in an elementary strips y varies from y = 0 to y = x while x varies from x = 0 to x = 1

9. dig dx dy

10. Thus the given doudle integral can be expressed as the repeated integral

11. Define triple integrals0:- Let be a function of three independent variables x,y,z defined for all points in a finite closed three dimensional region V of space. Divide V into n sub- regions of volumes , k= 1,2,….,n. let us select an arbitrary point in each and form the sum let the number of sub-division become infinite in such a way that the maximum dimensions of each approaches to zero.

12. If under these conditions, the limit Exists, which is independent of the way in which the points are chosen, then this limit is called the triple integral of over the region V, written is defined by

13. Example of triple integrals:- (1). Evaluate: Sol:- let the given triple integral be denoted by . Then

14. (2). evaluate:- where the region of integration V is a cylinder,which is bounded by the following surfaces: z = 0, z = 1,x²+y² = 4 Sol:- form the adjoining figure it is evident that in the region of integration V, z varies from z = 0 to z = 1, y varies from y = to y = and x varies from x = -2 to x = 2

15. dig

16. hence

17. Is an even function,so On substituting so that

18. .

19. OR

20. Thank you