Test 4b solutions

Test 4b solutions - x yields a slice of length ( x ) = h-x...

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Solutions to the Fourth Test C1 I: a ) 1 6 ( x + 3) 6 - 3 5 ( x + 3) 5 + C b ) 3 7 ( x + 2) 7 / 3 - 3 2 ( x + 2) 3 / 4 + C c ) - 1 3 (1 - x 4 ) 3 / 4 + C d ) 2 3 ( x + 2) 3 / 2 + 4( x + 2) 1 / 2 | 1 0 = 2 3 (3 3 / 2 - 1) + 4(3 1 / 2 - 1) e ) 0 Note that this is an integral of an odd function!. II: a) The intersection point of the two curves is at x = 1 / 3. Hence the area is given by Z 1 / 3 0 ±r x 3 - 3 x 2 ² d x = ± 1 3 ² 3 b) Because of symmetry the centroid is on the y = x line. Further the x component of the centroid multiplied by the area of the region is given by the integral Z 1 / 3 0 x ±r x 3 - 3 x 2 ² d x = 3 20 ± 1 3 ² 3 Thus x = y = 3 20 III: Let us call the width w = 20, the length of the base b = 100 and the height h = 6 and the density ρ = 5000. Slicing the ramp horizontally at height
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Unformatted text preview: x yields a slice of length ( x ) = h-x h b using similar triangles. Thus the work is given by gbw Z h x h-x h d x = gbwh 2 6 = 6 10 8 J b) The minimal time is given by 8 10 4 s 1 which is about 22 . 2 hours. IV: a) Slicing parallel to the y axis leads to the integral 2 Z 1 x 2-x d x = 4 15 (8 2-7) b) Slicing parallel to the y axis again leads to the integral Z 1 (2-x )d x = 3 2 2...
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Test 4b solutions - x yields a slice of length ( x ) = h-x...

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