To determine To find: The average value of the function f ( x ) = x 2 e 2 x on the interval [ 0 , 2 ] . Explanation Formula used: The average value of the function f on the interval [ a , b ] is 1 b − a ∫ a b f ( x ) d x . Method used: Integration by parts: “If u and v are differentiable functions, then ∫ u d v = u v − ∫ v d u .” Calculation: The function is f ( x ) = x 2 e 2 x over interval [ 0 , 2 ] . Thus, the average value of the function is, Average value = 1 2 ∫ 0 2 x 2 e 2 x d x . Solve the above integral by the help of integration by parts. Let u = x 2 and d v = e 2 x d x . Differentiate u = x 2 as follows. d u d x = d ( x 2 ) d x d u d x = 2 x d u = 2 x d x Integrate d v = e 2 x as follows. ∫ d v = ∫ e 2 x d x v = e 2 x 2 Substitute these corresponding values in the formula ∫ u d v = u v − ∫ v d u and simplify

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Chapter 8.2, Problem 31E

To determine

To find: The average value of the function f(x)=x2e2x on the interval [0,2] .

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Chapter 8.2, Problem 30E

Chapter 8.2, Problem 32E

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Chapter 8 Solutions

EBK CALCULUS WITH APPLICATIONS, BRIEF V

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