Explanation Given information: The given function is f ( x ) = { 2.3 x 3 − 3.1 x 2 + 2.7 x + 3 if x < 1 − 1.7 x 2 + 2.3 x + 4.3 if x ≥ 1 . Calculation: From the given piecewise function it is conclude that for x < 1 the value of the function is, f ( x ) = 2.3 x 3 − 3.1 x 2 + 2.7 x + 3 For x ≥ 1 the value of the function is, f ( x ) = − 1.7 x 2 + 2.3 x + 4.3 Property 5 of integration states that if c is lies between a and b , then the integral of f over the interval [ a , b ] is expressed as shown below, ∫ a b f ( x ) d x = ∫ a c f ( x ) d x + ∫ c b f ( x ) d x The value 1 is lies between the interval [ − 1 , 2 ] . So, according to the Property 5 of integration the integral of f over the interval [ − 1 , 2 ] is written as the sum of the integral of f over the interval [ − 1 , 1 ] and the integral of f over the interval [ 1 , 2 ] . ∫ − 1 2 f ( x ) d x = ∫ − 1 1 f ( x ) d x + ∫ 1 2 f ( x ) d x = ∫ − 1 1 ( 2.3 x 3 − 3.1 x 2 + 2.7 x + 3 ) d x + ∫ 1 2 ( − 1.7 x 2 + 2.3 x + 4.3 ) d x (1) Follow the steps using graphing calculator to find the definite integral. First press “ON” button on graphical calculator, To find the value of ∫ − 1 1 ( 2.3 x 3 − 3.1 x 2 + 2.7 x + 3 ) d x press keystrokes MATH 9 key, enter the function f ( x ) = 2.3 x 3 − 3.1 x 2 + 2.7 x + 3 with its lower limit and upper limits press keystrokes 2 . 3 X ^ 3 − 3

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Chapter 6.5, Problem 1TE

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To determine: The definite integral ∫−12f(x)dx for the given function.

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Ch. 6.1 - What is an antiderivative? Give an example.Ch. 6.1 - Prob. 2CQ Ch. 6.1 - What is the difference between an antiderivative...Ch. 6.1 - Prob. 4CQ Ch. 6.1 - In Exercises 1-4, verify directly that F is an...Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E Ch. 6.1 - In Exercises 1-4, verify directly that F is an...Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E

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The definite integral ∫ − 1 2 f ( x ) d x for the given function.

Solution Summary: The author calculates the integral of f over the interval left[a,bright].

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To determine: The definite integral ∫−12f(x)dx for the given function.

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Chapter 6 Solutions