1. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals MULTIPLE INTEGRALS PROGRAMME 24

2. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Summation in two directions Double integrals Triple integrals Applications Alternative notation Determination of areas by multiple integrals Determination of volumes by multiple integrals

3. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Summation in two directions Double integrals Triple integrals Applications Alternative notation Determination of areas by multiple integrals Determination of volumes by multiple integrals

4. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Summation in two directions The area δa of the shaded portion 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. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Summation in two directions The area of the rectangle is then the sum of all such strips: If now, δy → 0 and δx → 0 the area of the rectangle is given as:

6. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Summation in two directions Now:

7. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Summation in two directions Working another way where, again, the area δa of the shaded portion 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. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Summation in two directions The area of the rectangle is then the sum of all such strips: If now, δy → 0 and δx → 0 the area of the rectangle is given as:

9. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Summation in two directions Now: So here, the order of integration does not matter

10. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Summation in two directions Double integrals Triple integrals Applications Alternative notation Determination of areas by multiple integrals Determination of volumes by multiple integrals

11. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Summation in two directions Double integrals Triple integrals Applications Alternative notation Determination of areas by multiple integrals Determination of volumes by multiple integrals

12. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Double integrals The expression: is called a double integral and indicates that f (x, y) is first integrated with respect to x and the result is then integrated with respect to y If 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.

13. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Summation in two directions Double integrals Triple integrals Applications Alternative notation Determination of areas by multiple integrals Determination of volumes by multiple integrals

14. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Summation in two directions Double integrals Triple integrals Applications Alternative notation Determination of areas by multiple integrals Determination of volumes by multiple integrals

15. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Triple integrals The expression: is called a triple integral and is evaluated by starting with the innermost integral and working outwards. If 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.

16. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Summation in two directions Double integrals Triple integrals Applications Alternative notation Determination of areas by multiple integrals Determination of volumes by multiple integrals

17. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Summation in two directions Double integrals Triple integrals Applications Alternative notation Determination of areas by multiple integrals Determination of volumes by multiple integrals

18. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Applications Example 1: To find the area bounded by the x-axis and the ordinate at x = 5.

19. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Example 2: To find the area enclosed by the curves and Applications

20. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals 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. Applications

21. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Summation in two directions Double integrals Triple integrals Applications Alternative notation Determination of areas by multiple integrals Determination of volumes by multiple integrals

22. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Summation in two directions Double integrals Triple integrals Applications Alternative notation Determination of areas by multiple integrals Determination of volumes by multiple integrals

23. STROUD Worked examples and exercises are in the text Programme 24: 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.

24. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Summation in two directions Double integrals Triple integrals Applications Alternative notation Determination of areas by multiple integrals Determination of volumes by multiple integrals

25. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Summation in two directions Double integrals Triple integrals Applications Alternative notation Determination of areas by multiple integrals Determination of volumes by multiple integrals

26. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Determination of areas by multiple integrals To find the area of the polar curve r = f (θ) between the radius vectors θ = θ 1 and θ = θ 2 it is noted that the area of an element is r.δr. δθ. So the area in question is:

27. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Summation in two directions Double integrals Triple integrals Applications Alternative notation Determination of areas by multiple integrals Determination of volumes by multiple integrals

28. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Summation in two directions Double integrals Triple integrals Applications Alternative notation Determination of areas by multiple integrals Determination of volumes by multiple integrals

29. STROUD Worked examples and exercises are in the text Programme 24: Multiple integrals Determination of volumes by multiple integrals The element of volume is: Giving the volume V as: That is:

30. STROUD Worked examples and exercises are in the text Programme 24: 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 = x 2 + y 2.

31. STROUD Worked examples and exercises are in the text Programme 24: 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