P25 - 1 2 x 2 d x = Z 6 x 2-6 x d x = Z 6 x 2-6 x 2 d x = 2...

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Project 25 MAC 2311 1. We continue to look at the parabolic arch formed by the function f ( x ) = 3 x - 1 2 x 2 on [ 0 , 6 ] . We would like to calculate its area using Riemann Sums of rectangles. Sketch the parabola below on [ 0 , 6 ] and shade the area beneath it. If we the interval [ 0 , 6 ] into n subintervals: Interval 1 , Interval 2 , . . . , Interval n , what is the width of each one? Show or explain why the right endpoint of Interval i is: 0 + i 6 n . If we approximate the area of the arch with n rectangles of equal width and by choosing x * i to be right endpoints, what is the area of the rectangle on Interval i (in terms of i and n )? What is the limit that represents the area of the arch? Area = lim n i =1 Evaluate this limit using the technique learned in class (the formulas for sums of squares, cubes, etc. are in the notes or book).

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Evaluate these definite integrals using ONLY the previous answer and properties of integrals:
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Unformatted text preview: 1 2 x 2 d x = Z 6 x 2-6 x d x = Z 6 x 2-6 x + 2 d x = 2. If f ( x ) is negative on [ a, b ] , then R b a f ( x ) d x is negative. Why? Weren’t deﬁnite integrals deﬁned to be areas? (Review your deﬁnitions.) Sketch the following piecewise-deﬁned function, and evaluate the given deﬁnite integral in terms of area. Shade the regions whose areas you use to evaluate the deﬁnite integral. f ( x ) = √ 16-x 2 x ≤ 4-2 x < x < 3-2 x ≥ 3 Z 4-4 f ( x ) d x = Suppose that f ( x ) is the velocity (in m/s) of an object at time x in minutes. What is the total distance traveled by the object on [-4 , 2 ] ? Should Z 4 f ( x ) d x represent the total distance traveled by the object on [ 0 , 4 ] ? Think about what what a negative velocity means. . ....
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