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# P26 - this deﬁnite integral f x = sin πx x< √ x x...

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Project 26 MAC 2311 1. Here we show that the Fundamental Theorem of Calculus (FTC) does what it claims with one example. Examine the function f ( t ) = t 2 . Fix an unspecified number x a (as shown on the axes below), and then draw and shade the region represented by the definite integral R x a t 2 d t . 0 a x Show that the integral R x a t 2 d t is represented by the limit lim n →∞ n X i =1 h a + ( x - a n ) i i 2 ( x - a n ) . Evaluate the limit above using limits and summation fomulas—NOT by using FTC ( x is just some unspecified fixed number—treat it like a constant in the sum). Is your answer an antiderivative for f ( x ) as FTC states it should be? Is it F ( x ) - F ( a ) where F ( t ) = 1 3 t 3 ? (Expand and combine like terms to if needed.) So what is d dx Z x a t 2 d t ?

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2. Sketch the following piecewise-defined function, and evaluate the given definite integral us- ing FTC and properties of the definite integral.
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Unformatted text preview: this deﬁnite integral. f ( x ) = sin( πx ) x < √ x x ≥ Z 1-1 f ( x ) d x = Is your answer positive or negative? So, what has more area: one “hump” of the function sin( πx ) , or the area under the square root function from to 1 ? How much bigger is it? Just looking at your picture and thinking in terms of areas, is the function R x-1 f ( t ) d t increas-ing or decreasing at x =-1 2 ? at x = 0 ? at x = 1 2 ? Is it changing more rapidly at x = 1 2 or at x = 1 ? 3. Evaluate the integrals: Z 1 e t-e-t e t d t Z 9 1 2-√ ω w 2 d ω Which integral is the same as the area under the curve on its given interval?...
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