The Mean Value Theorem for Integrals tells us: {{CMD_194}} {{CMD_195}}The average value of a continuous function is in the range of the function.{{CMD_196}} {{CMD_197}} {{CMD_198}} Define an accumulation (area) function, {{MATH_199}}, {{MATH_200}} Since {{MATH_201}} is continuous on the interval {{MATH_202}} and differentiable on the interval {{MATH_203}}, we can apply the Mean Value Theorem to the function {{MATH_204}} on the interval {{MATH_205}}. Therefore, there exist a number {{MATH_206}} in {{MATH_207}} such that {{MATH_208}} But we know that {{MATH_209}}, and that {{MATH_210}}. Therefore, {{MATH_211}} {{CMD_212}} We demonstrate the principles involved in this version of the Mean Value Theorem in the following example.