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, 2≤u≤6, for which ∫62x2dx=f(u)(6−2)=4u2f(u)=u2
We integrate to obtain ∫62x2dx=[x33]62=13(216−8)=2083
Then 2083=4u2u2=5232≤u≤6u=√523≈4.163Disregard the negative solution since u>0.