For a function that is continuous on the closed interval

MATHEMATICS · Single correct · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

For a function ff that is continuous on the closed interval [a,b][a,b], which of the following equations correctly matches the statement guaranteed by the Mean Value Theorem for Integrals? (a) ∫abf(x) dx=f(a)+f(b)2 (b−a)\displaystyle \int_a^b f(x)\,dx = \dfrac{f(a)+f(b)}{2}\,(b-a) (b) ∫abf(x) dx=f ⁣(a+b2)(b−a)\displaystyle \int_a^b f(x)\,dx = f\!\left(\dfrac{a+b}{2}\right)(b-a) (c) ∫abf(x) dx=f(c) (b−a)\displaystyle \int_a^b f(x)\,dx = f(c)\,(b-a) (d) ∫abf(x) dx=f(a)+4f ⁣(a+b2)+f(b)6 (b−a)\displaystyle \int_a^b f(x)\,dx = \dfrac{f(a)+4f\!\left(\dfrac{a+b}{2}\right)+f(b)}{6}\,(b-a)

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