# HW10sol - of the sphere and x-axis points downwards(through...

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M408C, Homework 10 — Solution Instructor: Guoliang Wu 1. Find the area of the region enclosed by the curves y = x + 3 , y = ( x + 3) / 2 . Solution: To ﬁnd the intersections, ( y = x + 3 y = ( x + 3) / 4 x + 3 = ( x + 3) / 2 4( x + 3) = ( x + 3) 2 ( x + 3)( x - 1) = 0 x = - 3 or 1 . Thus, the area of the region enclosed by the curves is A = Z 1 - 3 ( x + 3 - ( x + 3) / 2) dx ( u = x + 3 ,du = dx ) = Z 4 0 ( u - u/ 2) du = 2 3 u 3 / 2 - u 2 4 # 4 0 = 16 3 - 4 = 4 3 1

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2. Find the volume of the cap of a sphere with radius r and height h . Solution: We choose the origin of the coordinate system to be the top
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Unformatted text preview: of the sphere, and x-axis points downwards (through the center of the sphere). The the “cap” is between x = 0 and x = h . The cross-section at r h level x is a circle, with radius p r 2-( r-x ) 2 . Thus the volume of the “cap” is V = Z h A ( x ) dx = Z h π ( r 2-( r-x ) 2 ) dx = π Z h (2 rx-x 2 ) dx = π ± rx 2-x 3 3 ² # h = π ( rh 2-h 3 3 ) 2...
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HW10sol - of the sphere and x-axis points downwards(through...

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