Mean Value Theorem for Integrals

• Let $a,b\in R$
• Let $a<b$
• If $f$ is cont on $[a,b]$ Then $\exists c\in (a,b)$ s.t $f(c)=\frac{1}{b-a}{\int }_a^{b}f(x)dx$
• Or $\exists c\in (a,b)$ s.t $\frac{F(b)-F(a)}{b-a}=f(c)$
• Kathleen likes this form: $\exists c\in [a,b]$ s.t ${\int }_a^{b}f(x)dx=f(c)(b-a)$

Alternate Area Theorem

• If $f:R\to \mathbb{R}$ is Continuous
• If $R$ is Simple Region
• Then, there is a point $(x_0,y_0)\in R$ s.t

$$\int {\int }_{R}f(x,y)dA=f(x_0,y_0)Area(R)$$

Intuition