Sample Third Exam: Calculus 1 (MAC 2311) Be sure to show all work and justify your answers; otherwise you will lose points. NO CALCULATORS OR NOTES. WRITE YOUR NAME ON THIS TEST! 1. Evaluate the following integrals (5 points each): (a) R x2 + 2x + 1 dx = (c) R √1 x (e) R 2 (1−2x)3 (g) R 2x cos( x7 ) dx = + 1 x + 1 x2 R (b) cos2 x sin x dx = dx = dx = (f) 2 ex 12x2 R 1 1+36x2 R 1 sin t+1 dt (h) 1 1 R (d) dx = dx = 1.(continued) Evaluate the following integrals ( 5 points each): R (i) cos3 y dy = (j) R tan x dx = √ 2. Approximate the value of 3 by applying Newton’s method to the function f (x) = x2 − 3. Start with x0 = 1 and calculate x2 . (15 points) 2 3. (i) Approximate the area under the curve y = x3 between the lines x = 0 and x = 2 by using the sum of the areas under 3 rectangles of equal width and height equal to x3 where x is the position of the right side of each rectangle. (5 points) (ii) (b) Use the formula n X k=1 k3 = (n)(n+1) 2 to compute an expression giving 2 the value for the total area under the sum of n rectangles of width n2 and height equal to x3 where x is the position of the right side of each rectangle between x = 0 and x = 2 and show what the limit as n → ∞ is. (10 points) 3 4. What is the average value of the function f (x) = sin x between 0 and (10 points) 4 π 2? 5. (a.) Evaluate the following integrals (5 points each): R3 2 4 − x2 dx = R3 (b.) |4 − x2 | dx = 1 (notice the lower limit is different in (b.) from that in (a.)) 5

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