Math 152 WI10 Exam 1 Review with Solutions

Math 152 WI10 Exam 1 Review with Solutions - MSLC Math 152...

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MSLC – Math 152 Exam 1 Review 1) Verify the following integrals give the indicated result. a. () ( ) ( ) ( ) 11 22 2 2 2 1 2 2 2 2 2 2 10 5 10 10 5 10 10 2 5 5 10 (5 ) 510 55 5 1 0 10 10 (5 ) (5 ) 10 25 5 50 5 25 1 25 10 25 10 10 dx x x c x xx x fx c x x x x x x x x x x x xx x xx =− + + −− −−+ = b. 2 2 c c ee dx f xe e c e e + += −+ = + =+ + = c. 2 2 2 ln 1 ln 1 1 () l n 1 l n 1 ( 1 ) ( 1 )2 ( 1 ) ( 1 ) 1 dx x x C x x x C x x x + + −− + =−= = +− + d. 2 21 (1 ) 1 1 1 )(1 ) 2 ) ) x x x dx C e C e e ++ + ==
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2) Consider the function 2 2 x −− . Use geometry to compute the value of 2 2 0 2 xdx . This integral is the area of a quarter of a circle with radius 2 r = . Area of a circle is 2 Ar π = so the area of a quarter circle is 2 11 1 44 4 2 2 ππ == = 3) a) Express the following sum in the form 1 n i a = . 333 3 3 1 55 1 05 1 5 5 21 2 1 2 1 2 1 n i n nn n n n n n n i = ⎡⎤ ⎛⎞ ⎛ ⎛⎞ ++ + + + + ⎢⎥ ⎜⎟ ⎜ ⎜⎟ ⎝⎠ ⎝ ⎝⎠ ⎣⎦ " b) The above is a right Riemann Sum for the definite integral () 2 b fxdx . Find and f(x) . 3 3 ( ) 22 1 5 25 7 ii i faixx i xaa i x so f x x ba x b b = + Δ Δ Δ= = +Δ=+ =+ = = = ∑∑ c) Find the value of the above sum as →∞ . 7 4 7 3 1 4 2 2 72 ( 1) 7 2 601 4 x xd x x +=+=+ −+ = d) The above sum can also be a right Riemann Sum for the definite integral 0 . Find the new function and bounds and compute the value of the sum as . 5 4 5 3 1 4 0 0 (2 ) 7 2 (( 2) 7 2 601 4 x x x + = + = + +=
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4) Express the definite integral 3 2 1 xdx as a right Riemann Sum and compute the sum using sigma algebra and the limit as n →∞ . Verify that the sum you compute is the same as computing the integral using the Fundamental Theorem of Calculus.
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Math 152 WI10 Exam 1 Review with Solutions - MSLC Math 152...

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