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Multiple Choice
What is the average value of the function over the interval ?
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Verified step by step guidance
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Step 1: Recall the formula for the average value of a function f(x) over an interval [a, b], which is given by: . Here, the interval is [1, 3] and the function is f(x) = x².
Step 2: Substitute the values of a = 1, b = 3, and f(x) = x² into the formula: . Simplify the denominator to get .
Step 3: Compute the definite integral . The antiderivative of x² is . Apply the Fundamental Theorem of Calculus: .
Step 4: Simplify the expression for the definite integral: . Thus, .
Step 5: Multiply the result of the integral by to find the average value: . The average value of the function f(x) = x² over the interval [1, 3] is .