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test3-0708

# test3-0708 - Department of Mathematics University of...

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Department of Mathematics, University of Toronto Term Test 3 – March 12, 2008 MAT 137Y, Calculus! Time Alloted: 1 hour 50 minutes Examiners: G. Amir, B. Khesin, R. Stanczak, B. Stephens, J. Sylvestre, S. Uppal 1. Evaluate the following integrals, and simplify your answer. (8%) (i) Z ( x + 1 ) 3 x 2 dx . (8%) (ii) Z e 2 x 1 + e 4 x dx . (10%) (iii) Z x 2 ( x 2 + 64 ) 3 / 2 dx . (10%) (iv) Z ln ( x 2 - 9 ) dx . (12%) 2. Sketch the region bounded by the curves y 2 = x and x = 2 y and find the volume of the solid by rotating the region about the y -axis. 3. (8%) (i) Suppose f is differentiable on ( a , b ) and continuous on [ a , b ] . Is it always true that d dx Z x a f ( t ) dt = Z x a d dt f ( t ) dt for x ( a , b ) ? If so, justify your answer with a proof. If the statement is false, what additional condition must be added so that the statement is true? (8%) (ii) Evaluate lim x 0 1 x 3 Z 4 x 0 arctan ( t 2 ) dt . 4. Consider the region under the curve

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