a. Explanation Given: The function f ( x ) = ( 2 x − 1 ) 2 − 4 , and the interval [ − 2 , 4 ] . Concept Used: The signed area between the graph of f and the x -axis on the interval [ a , b ] is given by the definite integral ∫ a b f ( x ) d x . 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. Power rule of integration: ∫ x n d x = x n + 1 n + 1 Calculation: In order tofind the exact value of the signed area of the region between the graph of f ( x ) = ( 2 x − 1 ) 2 − 4 and the x -axis on the interval [ − 2 , 4 ] , first plot the graph of the function f ( x ) = ( 2 x − 1 ) 2 − 4 on the interval [ − 2 , 4 ] . And check whether the area bounded by the graph is above the x -axis or below the x -axis. If the region bounded is above the x -axis, then the area will be positive and if it is below x -axis then the area would be negative. Graph of the function f ( x ) = ( 2 x − 1 ) 2 − 4 on the interval [ − 2 , 4 ] is shown below: From the above graph it is clear that the regions A and C are above the x -axis and hence positive. And the region B is below the x -axis and hence negative. Since the signed area between the graph of f and the x -axis on the interval [ a , b ] is given by the definite integral ∫ a b f ( x ) d x , so using this concept the signed area between the graph of f ( x ) = ( 2 x − 1 ) 2 − 4 and the x -axis on the interval [ − 2 , 4 ] is given by ∫ − 2 4 ( ( 2 x − 1 ) 2 − 4 ) d x = ∫ − 2 4 ( ( 4 x 2 − 4 x + 1 ) − 4 ) d x = ∫ − 2 4 ( 4 x 2 − 4 x − 3 ) d x = ∫ − 2 − 1 / 2 ( 4 x 2 − 4 x − 3 ) d x + ∫ − 1 / 2 3 / 2 ( 4 x 2 − 4 x − 3 ) d x + ∫ 3 / 2 4 ( 4 x 2 − 4 x − 3 ) d x = [ 4 3 x 3 − 2 x 2 − 3 x ] − 2 − 1 / 2 + [ 4 3 x 3 − 2 x 2 − 3 x ] − 1 / 2 3 / 2 + [ 4 3 x 3 − 2 x 2 − 3 x ] 3 / 2 4 = [ 4 3 ( ( − 1 2 ) 3 − ( − 2 ) 3 ) − 2 ( ( − 1 2 ) 2 − ( − 2 ) 2 ) − 3 ( ( − 1 2 ) − ( − 2 ) ) ] + [ 4 3 ( ( 3 2 ) 3 − ( − 1 2 ) 3 ) − 2 ( ( 3 2 ) 2 − ( − b. To determine The exact value of the absolute area of the region between the graph of f ( x ) = ( 2 x − 1 ) 2 − 4 and the x -axis on the interval [ − 2 , 4 ] using definite integrals and the Fundamental Theorem of Calculus.

14th Edition

ISBN: 9781464151088

Author: Associate Professor Laura Taalman, Peter Kohn

Publisher: W.H. Freeman & Company

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Chapter 4.6, Problem 33S

To determine

The exact value of the signed area of the region between the graph of f(x)=(2x−1)2−4 and the x -axis on the interval [−2, 4] using definite integrals and the Fundamental Theorem of Calculus.

To determine

The exact value of the absolute area of the region between the graph of f(x)=(2x−1)2−4 and the x -axis on the interval [−2, 4] using definite integrals and the Fundamental Theorem of Calculus.

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Chapter 4.6, Problem 32S

Chapter 4.6, Problem 34S

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The exact value of the signed area of the region between the graph of f ( x ) = ( 2 x − 1 ) 2 − 4 and the x -axis on the interval [ − 2 , 4 ] using definite integrals and the Fundamental Theorem of Calculus.

Solution Summary: The author calculates the exact value of the signed area between the graph of f and the x -axis on the interval left[a,bright].

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The exact value of the signed area of the region between the graph of f(x)=(2x−1)2−4 and the x -axis on the interval [−2, 4] using definite integrals and the Fundamental Theorem of Calculus.

The exact value of the absolute area of the region between the graph of f(x)=(2x−1)2−4 and the x -axis on the interval [−2, 4] using definite integrals and the Fundamental Theorem of Calculus.

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Chapter 4 Solutions