a. Explanation Given: The function f ( x ) = x 2 , and the interval [ − 2 , 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 ) = x 2 and the x -axis on the interval [ − 2 , 2 ] , first plot the graph of the function f ( x ) = x 2 on the interval [ − 2 , 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 ) = x 2 on the interval [ − 2 , 2 ] is shown below: From the above graph it is clear that both the regions A and B areabove the x -axis and hence the area bounded by the graph on the given interval is positive b. To determine The exact value of the absolute area of the region between the graph of f ( x ) = x 2 and the x -axis on the interval [ − 2 , 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 26S

To determine

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

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Chapter 4.6, Problem 25S

Chapter 4.6, Problem 27S

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

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

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