practice_prelim1_fa06 - 5 Find the length of the curve...

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Practice Prelim 1, Math 191, Fall 2006 No calculators. Show your work. Clearly mark each answer. This practice prelim may be longer than the actual prelim. 1. The max-min inequality for definite integrals is helpful in the following. [10] (a) Find upper and lower bounds for 2 0 x 1 + x 2 dx . (b) Rewrite the integral of part (a) by using the substitution u = x 2 . Find upper and lower bounds for the resulting integral. 2. Write the sum n k =1 n 3 + k 3 n 4 as a Riemann sum in order to evaluate lim n →∞ n k =1 n 3 + k 3 n 4 . [10] 3. Evaluate: [20] (a) 1 - 1 t cos(sin t ) dt (b) d dy 4 y 0 sin t cos t dt (c) 3 v - 1 dv (d) t 1 - 4 t 2 dt 4. Consider the region described by y 1 x 2 , y 8 x , and y x 2 16 . [15] (a) Compute its area. (b) Find its average height.
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Unformatted text preview: 5. Find the length of the curve defined by [10] y = Z x 2 1 1 2 r 1-1 t dt , 2 ≤ x ≤ 3 . 6. Solve the following. [20] (a) Find the volume of the solid generated by revolving the region enclosed by y = x 2 + 1 and y = x + 3 about x = 3 . (b) Find the volume of the solid generated by revolving the region bounded by y = 1-( x-1) 2 and the x-axis about the x-axis. 7. Find the surface area of the spherical cap obtained by rotating around the x-axis the curve described [15] parametrically by y = a sin θ , x = a cos θ , ≤ θ ≤ π/ 6 (for any constant a > )....
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