PROGRAMME 23

1. PROGRAMME 23 MULTIPLE INTEGRALS

2. Programme 23: Multiple integrals Summation in two directions Double integrals Triple integrals Applications Alternative notation Area of a plane figure bounded by a polar curve Determination of volumes by multiple integrals

3. Programme 23: Multiple integrals Summation in two directions Double integrals Triple integrals Applications Alternative notation Area of a plane figure bounded by a polar curve Determination of volumes by multiple integrals

4. Programme 23: Multiple integrals Summation in two directions The elemental area a of the rectangle bounded by the lines x = r, x = s, y = k and y = m is given as: The area of the vertical strip PQ is then:

5. Programme 23: Multiple integrals Summation in two directions The area of the rectangle is then the sum of all such strips: If now, y→ 0 and x → 0 the area of the rectangle is given as:

6. Programme 23: Multiple integrals Summation in two directions Now:

7. Programme 23: Multiple integrals Summation in two directions Working another way where, again, the elemental area a of the rectangle bounded by the lines x = r, x = s, y = k and y = m is given as: The area of the horizontal strip CD is then:

8. Programme 23: Multiple integrals Summation in two directions The area of the rectangle is then the sum of all such strips: If now, y→ 0 and x → 0 the area of the rectangle is given as:

9. Programme 23: Multiple integrals Summation in two directions Now: So here, the order of integration does not matter

10. Programme 23: Multiple integrals Summation in two directions Double integrals Triple integrals Applications Alternative notation Area of a plane figure bounded by a polar curve Determination of volumes by multiple integrals

11. Programme 23: Multiple integrals Double integrals The expression: is called a double integral and provided the four limits on the integral are all constant the order in which the integrations are performed does not matter. If the limits on one of the integrals involve the other variable then the order in which the integrations are performed is crucial.

12. Programme 23: Multiple integrals Summation in two directions Double integrals Triple integrals Applications Alternative notation Area of a plane figure bounded by a polar curve Determination of volumes by multiple integrals

13. Programme 23: Multiple integrals Triple integrals The expression: is called a triple integral and provided the six limits on the integral are all constant the order in which the integrations are performed does not matter. If the limits on the integrals involve some of the variables then the order in which the integrations are performed is crucial.

14. Programme 23: Multiple integrals Summation in two directions Double integrals Triple integrals Applications Alternative notation Area of a plane figure bounded by a polar curve Determination of volumes by multiple integrals

15. Programme 23: Multiple integrals Applications Example 1: To find the area bounded by the x-axis and the ordinate at x = 5.

16. Programme 23: Multiple integrals Applications Example 2: To find the area enclosed by the curves and

17. Programme 23: Multiple integrals Applications Example 3: Find the second moment of area of a rectangle 6 cm × 4 cm about an axis through one corner perpendicular to the plane of the figure.

18. Programme 23: Multiple integrals Summation in two directions Double integrals Triple integrals Applications Alternative notation Area of a plane figure bounded by a polar curve Determination of volumes by multiple integrals

19. Programme 23: Multiple integrals Alternative notation Sometimes double integrals are written in a different way. For example, the integral: could have been written as: Here the working starts from the right-hand side integral.

20. Programme 23: Multiple integrals Summation in two directions Double integrals Triple integrals Applications Alternative notation Area of a plane figure bounded by a polar curve Determination of volumes by multiple integrals

21. Programme 23: Multiple integrals Area of a plane figure bounded by a polar curve To find the area of the polar curve r = f ( ) between the radius vectors  = 1 and  = 2 it is noted that the area of an element is r.r.  . So the area in question is:

22. Programme 23: Multiple integrals Summation in two directions Double integrals Triple integrals Applications Alternative notation Area of a plane figure bounded by a polar curve Determination of volumes by multiple integrals

23. Programme 23: Multiple integrals Determination of volumes by multiple integrals The element of volume is: Giving the volume V as: That is:

24. Programme 23: Multiple integrals Determination of volumes by multiple integrals Example: Find the volume of the solid bounded by the planes z = 0, x = 1, x = 2, y = −1, y = 1 and the surface z = x2 + y2.

25. Programme 23: Multiple integrals Learning outcomes • Determine the area of a rectangle using a double integral • Evaluate double integrals over general areas • Evaluate triple integrals over general volumes • Apply double integrals to find areas and second moments • Apply triple integrals to find volumes