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Calculus II                                                          Quiz #2                                             February 24, 2004

Name____________________                   R.  Hammack                                                 Score ______
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(1)  Underoverscript[∑ , k = 1, arg3] (3 + 2k) =  Underoverscript[∑ , k = 1, arg3] 3 + Underoverscript[∑ , k = 1, arg3] 2k = Underov ... = 1, arg3] 3 + Underoverscript[2∑ , k = 1, arg3] k = (100) (3) + 2 (100 (100 + 1))/2 = 10400


(2)  Write the definite integral   ∫_0^1x/(x + 1) dx   as a limit of Riemann sums. (Just write the limit - don't try to find its value.)

∫_0^1x/(x + 1) dx = Underscript[lim , n∞] (Underoverscript[∑ , k = 1, arg3]   x_k^*/(x_k^* + 1) Δx)    OR   ∫_0^1x/(x + 1) dx = Underscript[lim , max (Δx) _i 0] (Underoverscript[∑ , k = 1, arg3]   x_k^*/(x_k^* + 1) (Δx) _i)  

Although either expression would count as a correct answer, further simplification is possible. If we set   Δx = (1 - 0)/n = 1/n, and choose as sample points the values x_k^* = k Δx = k/n, then the first expression simplifies as

∫_0^1x/(x + 1) dx = Underscript[lim , n∞] (Underoverscript[∑ , k = 1 ... rscript[lim , n∞] (Underoverscript[∑ , k = 1, arg3]   k/(k n + n^2))


(3) Find the value of the following definite integral by analyzing area under the graph.

∫_ (-1)^2 | 2x - 2 | dx = 5

Because the area under the graph between -1 and 2 consists of 2 triangles,
one of area (1/2)(2)(4) =4, and the other of area (1/2)(1)(2) =1.
The total area is thus 5.
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