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EXAMPLE 7Using the Mean Value Theorem for Integrals

Find the number(s) u guaranteed by the Mean Value Theorem for Integrals for 62x2dx.

Solution The Mean Value Theorem for Integrals states there is a number u, 2u6, for which 62x2dx=f(u)(62)=4u2f(u)=u2

We integrate to obtain 62x2dx=[x33]62=13(2168)=2083

Then 2083=4u2u2=5232u6u=5234.163Disregard the negative solution since u>0.