lecture31 - Use the Fundamental Theorem of Calculus to...

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§ 6.3 Definite Integrals and the Fundamental Theorem Math 121 Lecture 31 ! The Definite Integral ! Calculating Definite Integrals ! The Fundamental Theorem of Calculus ! Area Under a Curve as an Antiderivative
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! x = ( b a )/ n , x 1 , x 2 , …., x n are selected points from a partition [ a , b ]. Calculate the following integral. The figure shows the graph of the function f ( x ) = x + 0.5. Since f ( x ) is nonnegative for 0 " x " 1, the definite integral of f ( x ) equals the area of the shaded region in the figure below. 1 0.5 1
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The region consists of a rectangle and a triangle. By geometry, Thus the area under the graph is 0.5 + 0.5 = 1, and hence
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Calculate the following integral. The figure shows the graph of the function f ( x ) = x on the interval -1 " x " 1. The area of the triangle above the x -axis is 0.5 and the area of the triangle below the x -axis is 0.5. Therefore, from geometry we find that
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Unformatted text preview: Use the Fundamental Theorem of Calculus to calculate the following integral. An antiderivative of 3 x 1/3 – 1 – e 0.5 x is . Therefore, by the fundamental theorem, ( Heat Diffusion ) Some food is placed in a freezer. After t hours the temperature of the food is dropping at the rate of r ( t ) degrees Fahrenheit per hour, where (a) Compute the area under the graph of y = r ( t ) over the interval 0 " t " 2. (b) What does the area in part (a) represent? (a) To compute the area under the graph of y = r ( t ) over the interval 0 " t " 2, we evaluate the following. (b) Since the area under a graph can represent the amount of change in a quantity , the area in part (a) represents the amount of change in the temperature between hour t = 0 and hour t = 2. That change is 24.533 degrees Fahrenheit....
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lecture31 - Use the Fundamental Theorem of Calculus to...

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