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review sheet - P Vojta Math 1B Review Problems Fall 2009 1...

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Unformatted text preview: P. Vojta Math 1B Review Problems Fall 2009 1. Evaluate the following integrals (without using tables): (a) Z sin 4 x cos 3 xdx (d) Z 2 x 3 p 4- x 2 dx (g) Z dx (1 + x 2 ) 2 (b) Z 1 arcsin xdx (e) Z x sin 2 xdx (h) Z x 2 + 7 x- 6 x 3- 3 x 2 + 3 x + 7 dx (c) Z e x sin2 xdx (f) Z π/ 4 tan 2 θ sec 4 θ dθ (i) Z dx (1 + √ x ) 3 2. Evaluate Z x 2 p x 2- 4 xdx (tables allowed). 3. If possible, evaluate: (a) Z 1 ln xdx (b) Z ∞ 2 1 x 2- 1 dx (c) Z ∞ 1 x 2 x 3 + 2 dx 4. (a) Derive a reduction formula for R tan n xdx . (b) Use it to find R tan 6 xdx . 5. * Find Z sin 4 x cos 3 x dx . 6. Find the length of the arc y = x 3 / 2- 1, 0 ≤ x ≤ 4. 7. Find the area of (a) y = √ x , 0 ≤ x ≤ 1 rotated about the x-axis; (b) y = x 3 , 0 ≤ x ≤ 2 rotated about the y-axis. 8. Find the centroid of the region enclosed by the loop of the curve y 2 = x 3- x 4 . 9. Find the limits: (a) lim n →∞ ( n + 1) 5- ( n- 1) 5 n 4 , (b) lim n →∞ ( n !) 2 (2 n )! , (c) lim n →∞ (1 + n ) 1 /n ....
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review sheet - P Vojta Math 1B Review Problems Fall 2009 1...

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