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PROGRAMME 19

PROGRAMME 19. INTEGRATION APPLICATIONS 2. Programme 19: Integration applications 2. Volumes of solids of revolution Centroid of a plane figure Centre of gravity of a solid of revolution Lengths of curves Lengths of curves – parametric equations Surfaces of revolution

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PROGRAMME 19

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  1. PROGRAMME 19 INTEGRATION APPLICATIONS 2

  2. Programme 19: Integration applications 2 Volumes of solids of revolution Centroid of a plane figure Centre of gravity of a solid of revolution Lengths of curves Lengths of curves – parametric equations Surfaces of revolution Surfaces of revolution – parametric equations Rules of Pappus

  3. Programme 19: Integration applications 2 Volumes of solids of revolution Centroid of a plane figure Centre of gravity of a solid of revolution Lengths of curves Lengths of curves – parametric equations Surfaces of revolution Surfaces of revolution – parametric equations Rules of Pappus

  4. Programme 19: Integration applications 2 Volumes of solids 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. Programme 19: Integration applications 2 Volumes of solids of revolution To find the volume V of the solid of revolution consider a thin strip of the original plane figure with a volume V   y2.x

  6. Programme 19: Integration applications 2 Volumes of solids of revolution Dividing the whole plane figure up into a number of strips, each will contribute its own flat disc with volume V   y2.x

  7. Programme 19: Integration applications 2 Volumes of solids of revolution The total volume will then be: As x→ 0 the sum becomes the integral giving:

  8. Programme 19: Integration applications 2 Volumes of solids 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. Programme 19: Integration applications 2 Volumes of solids of revolution To find the volume V of the solid of revolution consider a thin strip of the original plane figure with a volume: V  area of cross section × circumference =2 xy.x

  10. Programme 19: Integration applications 2 Volumes of solids of revolution The total volume will then be: As x→ 0 the sum becomes the integral giving:

  11. Programme 19: Integration applications 2 Volumes of solids of revolution Centroid of a plane figure Centre of gravity of a solid of revolution Lengths of curves Lengths of curves – parametric equations Surfaces of revolution Surfaces of revolution – parametric equations Rules of Pappus

  12. Programme 19: 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.

  13. Programme 19: 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:

  14. Programme 19: Integration applications 2 Volumes of solids of revolution Centroid of a plane figure Centre of gravity of a solid of revolution Lengths of curves Lengths of curves – parametric equations Surfaces of revolution Surfaces of revolution – parametric equations Rules of Pappus

  15. Programme 19: 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:

  16. Programme 19: Integration applications 2 Volumes of solids of revolution Centroid of a plane figure Centre of gravity of a solid of revolution Lengths of curves Lengths of curves – parametric equations Surfaces of revolution Surfaces of revolution – parametric equations Rules of Pappus

  17. Programme 19: Integration applications 2 Lengths of curves To find the length of the arc of the curve y = f (x) between x = a and x = b let s be the length of a small element of arc so that:

  18. Programme 19: Integration applications 2 Lengths of curves In the limit as the arc length s approaches zero: and so:

  19. Programme 19: Integration applications 2 Volumes of solids of revolution Centroid of a plane figure Centre of gravity of a solid of revolution Lengths of curves Lengths of curves – parametric equations Surfaces of revolution Surfaces of revolution – parametric equations Rules of Pappus

  20. Programme 19: Integration applications 2 Lengths of curves – 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:

  21. Programme 19: Integration applications 2 Volumes of solids of revolution Centroid of a plane figure Centre of gravity of a solid of revolution Lengths of curves Lengths of curves – parametric equations Surfaces of revolution Surfaces of revolution – parametric equations Rules of Pappus

  22. Programme 19: Integration applications 2 Surfaces 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:

  23. Programme 19: Integration applications 2 Surfaces of revolution From previous work:

  24. Programme 19: Integration applications 2 Volumes of solids of revolution Centroid of a plane figure Centre of gravity of a solid of revolution Lengths of curves Lengths of curves – parametric equations Surfaces of revolution Surfaces of revolution – parametric equations Rules of Pappus

  25. Programme 19: Integration applications 2 Surfaces of revolution – parametric equations When the curve is defined by the parametric equations x = f () and y = F() then rotating a small arc s about the x-axis gives a thin band of area: Now: Therefore:

  26. Programme 19: Integration applications 2 Volumes of solids of revolution Centroid of a plane figure Centre of gravity of a solid of revolution Lengths of curves Lengths of curves – parametric equations Surfaces of revolution Surfaces of revolution – parametric equations Rules of Pappus

  27. Programme 19: Integration applications 2 Rules of Pappus • If 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 • If 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

  28. Programme 19: 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

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