Math.112HW3(2008-9,Spring)

Math.112HW3(2008-9,Spring) - a ≥ 0 the area of the region...

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“Due on March 2, 2009 up to 12:50 p.m.” Homework 3 for Math 112 1. Let f be a positive increasing continuous function on the interval [ a, b ] where a < b are positive real numbers. Show that Z b a π ( f ( x ) ) 2 dx + Z f ( b ) f ( a ) 2 πxf - 1 ( x ) dx = ( f ( b ) ) 2 - ( f ( a ) ) 2 , where f - 1 denotes the inverse of the invertible function f. ( Hint: Let R 1 be the region bounded by the curves x = a, x = b, y = 0 , and y = f ( x ) . Let R 2 be the region bounded by the curves y = f ( a ) , y = f ( b ) , x = 0 , and y = f ( x ) . Consider the volumes of the solids obtained by rotating about the x -axis the regions R 1 and R 2 . ) 2. Show that for α 0 , Z α 0 p 1 + 4 x 2 e - 2 x 2 dx q α 2 + (1 - e - α 2 ) 2 . ( Hint: Realize the integral in the left hand side of the inequality as the length of the arc of a curve, and realize the right hand side of the inequality as the length of a line segment. Use the fact that the smallest of the lengths of all the paths connecting two points is the length of the line segment connecting these points. ) 3. Let g ( x ) be a positive continuous function on [0 , ) . If for any
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Unformatted text preview: a ≥ 0 the area of the region bounded by the curves x = 0 , x = a, y = 0 , and y = g ( x ) is a 3 3 , what is the area of the surface generated by revolving the curve y = Z x g ( t ) dt, 1 ≤ x ≤ 2 about the x-axis? 4. Let f ( x ) be a positive continuously differentiable function on [1 , ∞ ] . For any a > 1 , let: A ( a ) = the area of the region bounded by the curves x = 1 , x = a, y = 0 , and y = f ( x ); L ( a ) = the length of the curve y = f ( x ) from x = 1 to x = a ; S ( a ) = the area of the surface obtained by rotating about the x-axis the curve y = f ( x ) from x = 1 to x = a ; V ( a ) = the volume of the solid obtained by rotating about the x-axis the region bounded by the curves x = 1 , x = a, y = 0 , and y = f ( x ) . Show that d 2 V da 2 dL da = d 2 A da 2 dS da . 1...
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This note was uploaded on 10/29/2010 for the course MATH 112 taught by Professor Ahmetguloglu during the Spring '10 term at Bilkent University.

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