sample_final - Problem 1 Evaluate the following...

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Problem 1. Evaluate the following integrals: (a) (4pts) Z x ln x dx = (b) (4pts) Z sin 10 θ cos θ dθ = (c) (4pts) Z 2 ( x + 4)( x + 6) dx = 2
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Problem 2. (15pts) Find the area of the region enclosed by the graphs of y = 4 x and y = 4 x 4 . 3
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Problem 3. Consider the function f ( x ) = x 1+ x 3 and the region R bounded by the graph of y = f ( x ) and the x -axis for 0 x < . (a) (6pts) Write an integral that represents the area of region R . Is this area finite? Justify your answer. (b) (6pts) Write an integral that represents the volume of the solid obtained by revolving region R about the x-axis . Is this volume finite? Justify your answer. (c) (6pts) Write an integral that represents the volume of the solid obtained by revolving region R about the y-axis . Is this volume finite? Justify your answer. 4
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Problem 4. (15pts) Find the length of the curve defined parametrically by x ( t ) = t 2 y ( t ) = 2 3 t 3 0 t 3 . 5
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Problem 5. Solve the following differential equations. Your answers should be in the form y = f ( x ).
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