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Math 1210 Quiz 8 March 20th, 2014 This is a take-home quiz. It contains 2 ordinary questions (35 points) and 1 extra credit question (15 points). The value of every question is indicated at the beginning of it. Please print this sheet and write your answers in the space provided. Name: UID: 1. (15 points) Compute the following indefinite integrals. R 2 2 dx (i) (5 points) (x √+1) x Hint: Expand the numerator first. (ii) (5 points) R √ 3y 2y 2 +5 dy R (iii) (5 points) sin x(1 + cos x)4 dx Hint: Use the generalized power rule. 2. (20 points) Find the area under the curve y = x3 + x between x = 0 and x = 1 by following the steps below: Hint: The answer is 43 . (i) (2 points) Subdivide the interval [0, 1] into n equal subintervals [x0 , x1 ], [x1 , x2 ], . . . , [xn−1 , xn ]. What is the length ∆x of every subinterval? For every k, write an expression for xk . (ii) (5 points) Sketch the rectangles that you will be using in the graph below and write down an expression for the area of the rectangle over [xk , xk+1 ]. Figure 1: Question 2 (iii) (5 points) Find the sum A(Rn ) of the areas of the n rectangles. 2 Pn n(n+1) 3 Hint: Remember that k=1 k = . 2 (iv) (4 points) The area under the curve will be the limit A = limn→∞ A(Rn ). Find this limit. Page 2 R (v) (4 points) Compute the indefinite integral F (x) = x3 + x dx (so that F 0 (x) = x3 + x) and find F (1) − F (0). You should be obtaining the same result as in (iv). You will understand why in section 4.4. 3. EXTRA CREDIT (15 points) (i) (10 points) Consider the differential equation dy = −y 2 x(x2 + 2)4 dx Use separation of variables in order to find the general solution of this equation. (ii) (5 points) Consider the differential equation dy x + 3x2 = dx y2 Show that y(x) = q 3 3 2 x 2 + 3x3 + 216 is a solution of this equation. Page 3