2018-Practice-Final-62-141.pdf - MATH 62-141-04 SOME...

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MATH 62-141-04 SOME PRACTICE QUESTIONS 1. (i) Let g ( x ) = Z ln x 0 e - 1+ t 2 dt . Find g 0 (1) and show your reason. (ii) Let f be a differentiable function such that f (0) = 0 and Z 1 0 f ( x ) f 0 ( x ) dx = 5. Find [ f (1)] 2 and show your reason. 2. Find the area of the region bounded by y = e - x , y = ln x , x = e , and x = 4. 3. Evaluate the following integrals. (i) Z 1 x (1 + ln x ) ln x dx . (ii) Z cot 3 x sin 6 x dx . (iii) Z sec θ tan 3 θ dθ . (iv) Z 1 x 2 x 2 - 1 dx .
4. Find the volume of the solid obtained by rotating about the x -axis the region bounded by y = 3 x , x = 0, and x = 1. 5. Use the method f cylindrical shells to find the volume of the solid obtained by rotating about the y -axis the region bounded by y = x 3 , y = 0, x = 1 and x = 2. 6. Determine whether the improper integrals are convergent. Show your reason. (i) Z 1 2 + sin x xe x dx . (ii) Z 1 0 1 x 2 + x dx . 7. Determine whether or not the sequence is convergent. Justify your answer.

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