Explanation Given: 0 < a < b are real numbers. Concept Used: The definite integral ∫ a b f ( x ) d x represents the area between the graph of f ( x ) and x -axis on the interval [ a , b ] . The area that is bounded by the graph of f ( x ) above the x -axis is positive. The area that is bounded by the graph of f ( x ) below the x -axis is negative. Area of trapezoid = 1 2 ( Sum of parallel sides ) × height Calculation: In order to prove that the definite integral ∫ a b x d x = 1 2 ( b 2 − a 2 ) , first plot the graph of the function f ( x ) = x on the interval [ a , b ] . And then break the region bounded by the graph and x -axis using some geometric shapes like triangles, rectangles or circles and find their area. Use the concept that the definite integral ∫ a b f ( x ) d x represents the area between the graph of f ( x ) and x -axis on the interval [ a , b ]

14th Edition

ISBN: 9781464140051

Author: TAALMAN, Laura, Kohn, Peter

Publisher: W. H. Freeman

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Chapter 4.3, Problem 59P

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To prove that ∫abx dx=12(b2−a2) for any real numbers 0<a<b , using geometry.

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Chapter 4.3, Problem 58P

Chapter 4.3, Problem 60P

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To prove that ∫ a b x d x = 1 2 ( b 2 − a 2 ) for any real numbers 0 < a < b , using geometry.