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MTH142_Assg08 - Assignment 8 This assignment is due at the...

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Assignment 8 This assignment is due at the beginning of class on Monday, May 5, 2008. 1. (a) For each definite integral below, use n = 10 rectangles to approximate the area under the curve. Please give answers to at least four decimal places. i. 2 1 1 x dx ii. 4 1 1 x dx iii. 8 1 1 x dx (b) What do you notice about your answers, above? (What is the pattern?) Use the observed pattern to guess at the value of 32 1 1 x dx. (c) Use the Fundamental Theorem of Calculus to explain the pattern you observed in part (b). 2. We know, from an earlier assignment, that the derivative of y = tan - 1 ( x ) is y = 1 1+ x 2 and so, according to the Fundamental Theorem of Calculus, 1 1 + x 2 dx = tan - 1 ( x ) + C. In this problem we solve the indefinite integral 1 1 + x 2 dx directly, using a certain substitution. Here is how: The expression 1 + x 2 should remind us of the Pythagorean Theorem. Draw a right triangle with legs of lengths 1 and x . (What is the length of the hypotenuse?) Let θ represent the angle opposite the leg of length x , so that tan( θ ) = x.
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