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Math1B Quiz4Solutions

# Math1B Quiz4Solutions - ∞ Z 1 1 3 √ 1 x 3 dx by...

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QUIZ 4 SOLUTION Question 1 (1) We will compare 1 3+ x 3 with 1 x 3 / 2 . 3 + x 3 x 3 , p 3 + x 3 x 3 / 2 , 1 3 + x 3 1 x 3 / 2 , The integral R 1 1 x 3 / 2 dx converges and so does R 1 1 3+ x 3 dx , by Comparison theorem. (2) Z 1 1 3 1 + x 3 dx. This integral behaves like 1 x for very large x but comparison 1 (1+ x 3 ) 1 / 3 1 x is useless but: 1 + x 3 2 x 3 , for x 1 , 3 p 1 + x 3 2 x 3 = 2 1 / 3 x. Thus 1 (1 + x 3 ) 1 / 3 1 2 1 / 3 x . The integral Z 1 1 2 1 / 3 x dx = 1 2 1 / 3 Z 1 1 x
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Unformatted text preview: ∞ Z 1 1 3 √ 1 + x 3 dx, by Comparison Theorem. Question 2 y = 3 x 1 / 2 L = 5 3 Z p 1 + y ( x ) 2 dx = 5 3 Z √ 1 + 9 xdx = 1 9 2 3 (1 + 9 x ) 3 / 2 ± ± ± 5 / 3 = = 2 27 (16 3 / 2-1 3 / 2 ) = 2 27 63 = 14 3 . 1...
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