Calculus 2 Practice Final Exam - Math 231 Fall 2009

1.	Evaluate the integrals a. ∫ [x^2/√(1 – x^2)] dx Hint: Use x = sin θ b. ∫ [x^3/√(1 + x^4)] dx Hint: Try a substitution Answer: a I = (1/2)sin^-1x - (1/2)x/√(1 - x^2) + C b. I = (1/6) ( 1 + x4 )^3/2 + C
2.	Do the following series converge or diverge? ∞ ∞ a. ∑ (1/n) √ ln n b. ∑ (2n – 1) / (n^3 + n + 3) n=2 n=1 a. Diverges b. Converges
3.	Let C be the curve with polar equation r = e^√3θ , 0 ≤ θ ≤ 2π a. Calculate the length of C. b. Find all points on C at which the tangent line is horizontal. Answers: a. (2/√3) [e^2/√3˗ 1] b. e√^3 (5π/6) , 5π/6) and e√^3 (11π/6) , 11π/6)
4.	Sketch the curve given by parametric equations x = sin t and y = cos t 0 ≤ t ≤ π/4. Indicate with an arrow the direction in which the curve is traced as t increases. Answer: Arc of a circle traced in clockwise direction
5.	Find the area of the region inside the polar curve r = 4 sin 3θ Answer: 4π Click here to continue with this exam.

Return to another Final Exam

Return to Table of Course Notes