**Mean Value Theorem for Integrals:** If \(f\) is continuous on \([a,b]\), then there exists a number \(c\) in \([a,b]\) such that \[\int_a^b {f\left( x \right)dx = f\left( c \right)\left( {b - a} \right)} \]

**Instructions:**Drag

*a*and

*b*to see c calculated dynamically. The area under the curve from

*a*to

*b*is the same as the area of the pink rectangle formed by

*b - a*and

*f(c)*.