Exam (elaborations)

Math 224 Exam 1 Questions & Answers Graded A.

Math 224 Exam 1 Questions & Answers Question 1. Consider the following integral: ∫ π cos2( ) sin3( ) 0 (a) (10 points): Evaluate the integral. x x dx. (b) (5 points): Which of the following expressions represents the integral as a Riemann sum with right endpoints? (Circle your answer) A. lim Σ πi cos2. π Σ sin3. π Σ B. lim nΣ−1 π cos2. πi Σ sin3. πi Σ D. lim Σ cos2. πi Σ sin3. πi Σ Question 2. (10 points): Let f be a continuous function. Compute the following derivatives: (a) d Σ∫ x2−3x f (t) dtΣ dx 5 (b) d Σ∫ x (3 2 8) Σ Question 3. Evaluate each of the following integrals. (a) (10 points): ∫ x√x 1 dx Solution: Let u = x 1 then du = dx and x = u − 1. Then, ∫ x√x 1 dx = ∫ (u − 1)√u du = ∫ .u3/2 − u1/2Σ du = 2 u5/2 − 2 u3/2 C 5 3 = 2 (x 1)5/2 − 2 (x 1)3/2 C. 5 3 (b) (10 points): ln x √x dx Solution: Use integration by parts as follows: u = ln x dv = x−1/2 dx du = 1 dx v = 2x1/2 x Therefore, ∫ ln x √ ∫ 1 1/2 = 2√x ln x − 2 ∫ x−1/2 dx = 2√x ln x − 4√x C.