hw4sol

# hw4sol - x –axis So we obtain the following Volume = π Z...

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Fall 2010, MAT21B, Solution to Written Homework 4 1. The region bounded by the curve y = 1 + cos x and the x - axis between 0 x 2 π is revolved about the x - axis to generate a solid. Find the volume of the solid. Solution: Let f ( x ) = 1 + cos x . We will use the disk method to calculate the volume. Therefore, Volume = Z 2 π 0 π ( f ( x )) 2 dx = π Z 2 π 0 ( 1 + 2 cos x + cos 2 x ) dx = π Z 2 π 0 1 dx + 2 π Z 2 π 0 cos xdx + π Z 2 π 0 cos 2 xdx Using the double angle formula cos 2 x = 1 + cos(2 x ) 2 and the fact that R 2 π 0 cos xdx = 0, we simplify the above expression: Volume = 2 π 2 + π Z 2 π 0 1 + cos(2 x ) 2 dx = 3 π 2 + π sin(2 x ) 4 ± ± ± ± 2 π 0 = 3 π 2 2. The region bounded above by the curve y = sec( x ), below by y = tan( x ), on the left by the line x = 0, on the right by the line x = 1 is revolved about the x - axis to generate a solid. Find the volume of the solid. Solution:Let f 1 ( x ) = sec x and let f 2 ( x ) = tan x . We will use the washer method to ﬁnd the volume of this solid of revolution about the
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Unformatted text preview: x –axis. So we obtain the following: Volume = π Z 1 ( f 1 ( x )) 2-( f 2 ( x )) 2 dx = π Z 1 sec 2 x-tan 2 xdx = π Z 1 1 dx = π where we used the fact that sec 2 x-tan 2 x = 1. 3. Find the volume of the solid generated by revolving about the y-axis the region bounded by the curves y = 3 x +22 √ x 3 +11 x 2 +4 , the x-axis, and the lines x = 0 and x = 1. Solution: Let f ( x ) = 3 x +22 √ x 3 +11 x 2 +4 and observe that the volume of the solid obtained by revolving around the y –axis is most easily obtained by the shell method. Hence, Volume = Z 1 2 πxf ( x ) dx = 2 π Z 1 3 x 2 + 22 x √ x 3 + 11 x 2 + 4 dx = 2 π h 2 √ x 3 + 11 x 2 + 4 i 1 = 4 π (4-2) = 8 π. 1...
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