11Breview6sol

11Breview6sol - econ 11b ucsc ams 11b Review Questions 6...

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econ 11b ucsc ams 11b Review Questions 6 Solutions 1. Compute the following definite integrals. Use the Fundamental Theorem of Calculus , (i.e. don’t use limits of right-hand sums). a. Z 4 0 3 xdx 3 x 2 + 1 = 9 4 ( x 2 + 1) 2 / 3 ± ± ± ± 4 0 = 9 4 ( 17 2 / 3 - 1 2 / 3 ) 12 . 62585. b. Z 3 1 ( x 2 +2 x )( x 3 +3 x 2 - 1) 3 dx = 1 12 ( x 3 + 3 x 2 - 1) 4 ± ± ± ± 3 1 = 1 12 ( 53 4 - 3 4 ) = 657533 . 333. c. Z e 2 e dx x ln x = ln | ln x | ± ± ± e 2 e = ln(ln e 2 ) - ln(ln e ) = ln 2 - ln 1 = ln 2. Note: Problems a., b. and c., above, involve the same integrals that appeared in 1abc of RQ 5. See the solutions of RQ 5 to see how the antiderivatives were found. d. Substitute u = - 0 . 05 t , du = - 0 . 05 dt dt = - 20 du , and also, change the limits of integration according to the substitution: t = 0 u = - (0 . 05) · 0 = 0 and t = 10 u = - (0 . 05) · 10 = - 0 . 5. Then, Z 10 0 1000 e - 0 . 05 t dt = Z - 0 . 5 0 - 20000 e u du = - 20000 e u ± ± ± - 0 . 5 0 = - 20000 ( e - 0 . 5 - e 0 ) 7869 . 387 . e. Substitute
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11Breview6sol - econ 11b ucsc ams 11b Review Questions 6...

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