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Calculus II Midterm Summer 2013 Name________________________________ Make sure you show all of your work neatly on this paper. Solutions without correct supporting work will not earn credit. Circle your answers. 1. Set up, but do not evaluate, the integrals to find the volumes of the solids generated by revolving the shaded region about the indicated lines. (11 points each) a. The x-axis b. x=7 2. Find the length of the arc from x = 0 to x = 2 on the curve y 3/ 2 2 2 x 1 . 3 (11 points) 3. Consider a spherical water tank whose radius is 10 ft and whose center is 50 ft above the ground. Set up, but do not evaluate, the integral to find the work required to fill this tank by pumping water up from ground level. Recall that the density of water is 62.4 lb/ft3. (11 points) 4. Evaluate: ∫ 𝑥 2 ln 𝑥 𝑑𝑥 5. Evaluate: ∫ 6. Evaluate: ∫ sec2 (ln 𝜃) 𝜃 √9𝑥 2 −25 𝑥3 (11 points) 𝑑𝜃 𝑑𝑥 , 𝑥 > (11 points) 5 3 (11 points) ∞ 7. Evaluate: ∫0 𝑒 −3𝑥 𝑑𝑥 8. Evaluate: ∫ (𝑥−6)(𝑥+2)2 𝑑𝑥 𝑥 (11 points) (11 points) 9. Solve: (sec 𝑥) 𝑦 ′ = 𝑦 3 , 𝑦(0) = 3 10. Evaluate: ∫ 𝑥 sinh 𝑥 𝑑𝑥 (11 points) (11 points) 11. a. Write the first 4 terms of the sequence defined by the recursive formula: 𝑎𝑛+1 = 𝑎𝑛 + (𝑛 − 1)! , 𝑎0 = 2 b. Find an explicit formula for the general nth term of the sequence: {1,-4,9,-16,25,…} . (4 points) (4 points) 12. Find the limit of the sequence or determine if the limit does not exist: 2 𝑛 {(1 + 𝑛) } (10 points) 13. Determine if the series converges or diverges. If it converges, find the sum. (11 points) ∑∞ 𝑘=1 2𝑘 3𝑘