7. Relate Riemann integrals: Prove the two theorems of Pappus: (a) The area of the surface...

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7. Relate Riemann integrals: Prove the two theorems of Pappus: (a) The area of the surface of revolution which is produced by a plane arc when it is rotated about an axis on the same plane with the arc which does not intersect the arc is equal to the product of the length of the arc times the circumference of the circle swept by its CM. (b) The volume of the solid of revolution which is produced by a plane figure when it is rotated about an axis on the same plane with the figure which does not intersect the figure is equal to the product of the area of the figure times the circumference of the circle swept by its CM. Comment: Sometimes these two theorems appear (incorrectly) in the literature as Guldin's theorems. Pappus was the last of the great ancient Greek geometers, c. 290-350 AD. Guldin was Swiss mathematician, b. 06/12/1577, d. 11/03/1643. These theorems were proved in Book VII of Pappus' Mathematical Collection. Guldin dupli- cated Pappus' results. (c) Apply the above theorems: find the CM of a semicircle; find the CM of a semidisc; find the area of a torus. 7. Relate Riemann integrals: Prove the two theorems of Pappus: (a) The area of the surface of revolution which is produced by a plane arc when it is rotated about an axis on the same plane with the arc which does not intersect the arc is equal to the product of the length of the arc times the circumference of the circle swept by its CM. (b) The volume of the solid of revolution which is produced by a plane figure when it is rotated about an axis on the same plane with the figure which does not intersect the figure is equal to the product of the area of the figure times the circumference of the circle swept by its CM. Comment: Sometimes these two theorems appear (incorrectly) in the literature as Guldin's theorems. Pappus was the last of the great ancient Greek geometers, c. 290-350 AD. Guldin was Swiss mathematician, b. 06/12/1577, d. 11/03/1643. These theorems were proved in Book VII of Pappus' Mathematical Collection. Guldin dupli- cated Pappus' results. (c) Apply the above theorems: find the CM of a semicircle; find the CM of a semidisc; find the area of a torus. 7. Relate Riemann integrals: Prove the two theorems of Pappus: (a) The area of the surface of revolution which is produced by a plane arc when it is rotated about an axis on the same plane with the arc which does not intersect the arc is equal to the product of the length of the arc times the circumference of the circle swept by its CM. (b) The volume of the solid of revolution which is produced by a plane figure when it is rotated about an axis on the same plane with the figure which does not intersect the figure is equal to the product of the area of the figure times the circumference of the circle swept by its CM. Comment: Sometimes these two theorems appear (incorrectly) in the literature as Guldin's theorems. Pappus was the last of the great ancient Greek geometers, c. 290-350 AD. Guldin was Swiss mathematician, b. 06/12/1577, d. 11/03/1643. These theorems were proved in Book VII of Pappus' Mathematical Collection. Guldin dupli- cated Pappus' results. (c) Apply the above theorems: find the CM of a semicircle; find the CM of a semidisc; find the area of a torus. 7. Relate Riemann integrals: Prove the two theorems of Pappus: (a) The area of the surface of revolution which is produced by a plane arc when it is rotated about an axis on the same plane with the arc which does not intersect the arc is equal to the product of the length of the arc times the circumference of the circle swept by its CM. (b) The volume of the solid of revolution which is produced by a plane figure when it is rotated about an axis on the same plane with the figure which does not intersect the figure is equal to the product of the area of the figure times the circumference of the circle swept by its CM. Comment: Sometimes these two theorems appear (incorrectly) in the literature as Guldin's theorems. Pappus was the last of the great ancient Greek geometers, c. 290-350 AD. Guldin was Swiss mathematician, b. 06/12/1577, d. 11/03/1643. These theorems were proved in Book VII of Pappus' Mathematical Collection. Guldin dupli- cated Pappus' results. (c) Apply the above theorems: find the CM of a semicircle; find the CM of a semidisc; find the area of a torus.

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The problem youve shared is asking to prove the two theorems of Pappus which relate to the creation of surfaces and solids of revolutions via integration These theorems are quite elegant as they provi... View the full answer