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PAPPUS THEOREM

Lesson 15 pappus theorem

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Page 1: Lesson 15 pappus theorem

PAPPUS THEOREM

Page 2: Lesson 15 pappus theorem
Page 3: Lesson 15 pappus theorem

rsS 2

Thus,

rotation of axis the toline

generating theofmidpoint thefrom distance

r2 C d

dx

dy1length arc s where

2

dx

sdS

centroid geometric scurve' by the travelleddistance theis d

Page 4: Lesson 15 pappus theorem

The following table summarizes the surface areas calculated using Pappus's centroid theorem for various surfaces of revolution.

Page 5: Lesson 15 pappus theorem
Page 6: Lesson 15 pappus theorem

centroid by the travelledncecircumfere theof radius

r 2 C d

region theof area A where

AdV

rAV 2

Thus,

centroid geometric slamina' by the travelleddistance theis d

Page 7: Lesson 15 pappus theorem

The following table summarizes the surface areas and volumes calculated using Pappus's centroid theorem for various solids and surfaces of revolution.

Page 8: Lesson 15 pappus theorem

THEOREM OF PAPPUSThe Theorem of Pappus states that when a region R is rotated about a line l, the volume of the solid generated is equal to the product of the area of R and the distance the centroid of the region has traveled in one full rotation.

section cross theof area A

section cross theof centroid the torevolution of axis thefrom distance r

where

2

rAV

Page 9: Lesson 15 pappus theorem

1. Find the volume bounded by a doughnut – shaped surface formed by rotating a circle of radius a about an axis whose distance from the circle’s center is b.

AdV 2)(aA )(2 bd

baV 22

baV 222

EXAMPLE

Page 10: Lesson 15 pappus theorem

2. Find the centroid of the semicircular area in the 1st and 4th quadrant bounded by using Pappus’ Theorem.

222 ayx

3. Find the volume of a circular cone of radius and height h using Pappu’s Theorem.

Page 11: Lesson 15 pappus theorem

4. Determine the amount of paint required to paint the inside and outside surfaces of the cone, if one gallon of paint covers

300 ft2 using Pappu’s Theorem.

Page 12: Lesson 15 pappus theorem

rotation of axis the toline

generating theofmidpoint thefrom distance r

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5. A right circular cone is generated when the region bounded by the line y = x and the vertical lines x = 0 and x = r is revolved about the x axis. a. Use the Theorem of Pappus to show that the volume of this cone is

b. Use the second Theorem of Pappus to determine the surface area of this region as well.

Verify this with the surface area formula for a cone.

3

3

1rV

6. Find the volume of the solid figure generated by revolving an equilateral triangle of side L about one of its sides. Use the Theorem of Pappus to determine the surface area of this region as well.

Page 19: Lesson 15 pappus theorem

7. Let R be the triangular region bounded by the line y = x, the x-axis, and the vertical line x = r. When R is rotated about the x-axis, it generates a cone of volume

Use the Theorem of Pappus to determine the y-coordinate of the centroid of R. Then use similar reasoning to find the x- coordinate of the centroid of R.

3

3

4rV

8. Find the volume of the solid figure generated when a square of side L is revolved about a line that is outside the square, parallel to two of its sides, and located s units from the closer side. Use thesecond Theorem of Pappus to determine the surface area of this region as well.

Page 20: Lesson 15 pappus theorem