Chap6_Sec2

Chap6_Sec2 - SPHERES Show that the volume of a sphere of...

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SPHERES Show that the volume of a sphere of radius r is If we place the sphere so that its center is at the origin, then the plane P x intersects the sphere in a circle whose radius, from the Pythagorean Theorem, is: So, the cross-sectional area is: Using the definition of volume with a = -r and b = r , we have: 2 2 y r x = - 3 4 3 . V r π = Example 1 (6.2) 2 2 2 ( ) ( ) A x y r x = = - ( ) 2 2 2 2 0 3 3 2 3 0 3 4 3 ( ) 2 ( ) 2 2 3 3 r r r r r r V A x dx r x dx r x dx x r r x r r - - = = - = - = - = - =
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Find the volume of the solid obtained by rotating about the x-axis the region under the curve from 0 to 1. Illustrate the definition of volume by sketching a typical approximating cylinder. The region is shown in the first figure. If we rotate about the x -axis, we get the solid shown in the next figure. s When we slice through the point x , we get a disk with radius x . The area of the cross-section is:
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Chap6_Sec2 - SPHERES Show that the volume of a sphere of...

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