a. Explanation Given: The function f ( x ) = 3 x 2 + 5 x − 2 , and the interval [ − 3 , 2 ] . 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 ) = 3 x 2 + 5 x − 2 and the x -axis on the interval [ − 3 , 2 ] , first plot the graph of the function f ( x ) = 3 x 2 + 5 x − 2 on the interval [ − 3 , 2 ] . 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 ) = 3 x 2 + 5 x − 2 on the interval [ − 3 , 2 ] 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 ) = 3 x 2 + 5 x − 2 and the x -axis on the interval [ − 3 , 2 ] is given by ∫ − 3 2 | 3 x 2 + 5 x − 2 | d x = ∫ − 3 − 2 ( 3 x 2 + 5 x − 2 ) d x + ∫ − 2 1 / 3 − ( 3 x 2 + 5 x − 2 ) d x + ∫ 1 / 3 2 ( 3 x 2 + 5 x − 2 ) d x = [ 3 3 x 3 + 5 2 x 2 − 2 x ] − 3 − 2 − [ 3 3 x 3 + 5 2 x 2 − 2 x ] − 2 1 / 3 + [ 3 3 x 3 + 5 2 x 2 − 2 x ] 1 / 3 2 = [ ( ( − 2 ) 3 − ( − 3 ) 3 ) + 5 2 ( ( − 2 ) 2 − ( − 3 ) 2 ) − 2 ( ( − 2 ) − ( − 3 ) ) ] − [ ( ( 1 3 ) 3 − ( − 2 ) 3 ) + 5 2 ( ( 1 3 ) 2 − ( − b. To determine The exact value of the absolute area of the region between the graph of f ( x ) = 3 x 2 + 5 x − 2 and the x -axis on the interval [ − 3 , 2 ] using definite integrals and the Fundamental Theorem of Calculus.

14th Edition

ISBN: 9781464140051

Author: TAALMAN, Laura, Kohn, Peter

Publisher: W. H. Freeman

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Chapter 4.6, Problem 31S

To determine

The exact value of the signed area of the region between the graph of f(x)=3x2+5x−2 and the x -axis on the interval [−3, 2] 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)=3x2+5x−2 and the x -axis on the interval [−3, 2] using definite integrals and the Fundamental Theorem of Calculus.

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Chapter 4.6, Problem 30S

Chapter 4.6, Problem 32S

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Chapter 4 Solutions

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The exact value of the signed area of the region between the graph of f ( x ) = 3 x 2 + 5 x − 2 and the x -axis on the interval [ − 3 , 2 ] 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(x) and the x -axis on the interval.

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The exact value of the signed area of the region between the graph of f(x)=3x2+5x−2 and the x -axis on the interval [−3, 2] 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)=3x2+5x−2 and the x -axis on the interval [−3, 2] using definite integrals and the Fundamental Theorem of Calculus.

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