1. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 INTEGRATION APPLICATIONS 2 PROGRAMME 20

2. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Volume of a solid of revolution Centroid of a plane figure Centre of gravity of a solid of revolution Length of a curve Surface of revolution Rules of Pappus

3. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Volume of a solid of revolution Centroid of a plane figure Centre of gravity of a solid of revolution Length of a curve Surface of revolution Rules of Pappus

4. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Volume of a solid of revolution If a plane figure bounded by the curve y = f (x), the x-axis and the ordinates x = a and x = b, rotates through a complete revolution about the x-axis, it will generate a solid symmetrical about Ox

5. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Volume of a solid of revolution To find the volume V of the solid of revolution consider a thin strip of the original plane figure with a volume δV = π y 2.δx

6. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Volume of a solid of revolution Dividing the whole plane figure up into a number of strips, each will contribute its own flat disc with volume δV = π y 2.δx

7. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Volume of a solid of revolution The total volume will then be: As δx → 0 the sum becomes the integral giving:

8. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Volume of a solid of revolution If a plane figure bounded by the curve y = f (x), the x-axis and the ordinates x = a and x = b, rotates through a complete revolution about the y-axis, it will generate a solid symmetrical about Oy

9. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 To find the volume V of the solid of revolution consider a thin strip of the original plane figure with a volume: δV = area of cross section × circumference =2π xy.δx Volume of a solid of revolution

10. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Volume of a solid of revolution The total volume will then be: As δx → 0 the sum becomes the integral giving:

11. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Volume of a solid of revolution Centroid of a plane figure Centre of gravity of a solid of revolution Length of a curve Surface of revolution Rules of Pappus

12. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Volume of a solid of revolution Centroid of a plane figure Centre of gravity of a solid of revolution Length of a curve Surface of revolution Rules of Pappus

13. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Centroid of a plane figure The coordinates of the centroid (centre of area) of a plane figure are obtained by taking the moment of an elementary strip about the coordinate axes and then summing over all such strips. Each sum is then approximately equal to the moment of the total area taken as acting at the centroid.

14. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Centroid of a plane figure In the limit as the width of the strips approach zero the sums are converted into integrals giving:

15. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Volume of a solid of revolution Centroid of a plane figure Centre of gravity of a solid of revolution Length of a curve Surface of revolution Rules of Pappus

16. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Volume of a solid of revolution Centroid of a plane figure Centre of gravity of a solid of revolution Length of a curve Surface of revolution Rules of Pappus

17. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Centre of gravity of a solid of revolution The coordinates of the centre of gravity of a solid of revolution are obtained by taking the moment of an elementary disc about the coordinate axis and then summing over all such discs. Each sum is then approximately equal to the moment of the total volume taken as acting at the centre of gravity. Again, as the disc thickness approaches zero the sums become integrals:

18. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Volume of a solid of revolution Centroid of a plane figure Centre of gravity of a solid of revolution Length of a curve Surface of revolution Rules of Pappus

19. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Volume of a solid of revolution Centroid of a plane figure Centre of gravity of a solid of revolution Length of a curve Surface of revolution Rules of Pappus

20. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Length of a curve To find the length of the arc of the curve y = f (x) between x = a and x = b let δs be the length of a small element of arc so that:

21. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 In the limit as the arc length δs approaches zero: and so: Length of a curve

22. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Length of a curve Parametric equations Instead of changing the variable of the integral as before when the curve is defined in terms of parametric equations, a special form of the result can be established which saves a deal of working when it is used. Let:

23. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Volume of a solid of revolution Centroid of a plane figure Centre of gravity of a solid of revolution Length of a curve Surface of revolution Rules of Pappus

24. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Volume of a solid of revolution Centroid of a plane figure Centre of gravity of a solid of revolution Length of a curve Surface of revolution Rules of Pappus

25. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Surface of revolution When the arc of a curve rotates about a coordinate axis it generates a surface. The area of a strip of that surface is given by:

26. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 From previous work: Surface of revolution

27. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Surface of revolution Parametric equations When the curve is defined by the parametric equations x = f (θ) and y = F(θ) then rotating a small arc δs about the x-axis gives a thin band of area: Now: Therefore:

28. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Volume of a solid of revolution Centroid of a plane figure Centre of gravity of a solid of revolution Length of a curve Surface of revolution Rules of Pappus

29. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Volume of a solid of revolution Centroid of a plane figure Centre of gravity of a solid of revolution Length of a curve Surface of revolution Rules of Pappus

30. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Rules of Pappus 1If an arc of a plane curve rotates about an axis in its plane, the area of the surface generated is equal to the length of the line multiplied by the distance travelled by its centroid 2If a plane figure rotates about an axis in its plane, the volume generated is equal to the area of the figure multiplied by the distance travelled by its centroid. Proviso: The axis of rotation must not cut the rotating arc or plane figure

31. STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 Learning outcomes Calculate volumes of revolution Locate the centroid of a plane figure Locate the centre of gravity of a solid of revolution Determine the lengths of curves Determine the lengths of curves given by parametric equations Calculate surfaces of revolution Calculate surfaces of revolution using parametric equations Use the two rules of Pappus