Math 22 The Area of a Region Bounded by Two Graphs

Area of a Region Bounded by Two Graphs

 If $f$ and $g$ are continuous on $[a,b]$ and $g(x)\leq f(x)$ for all $x$ in $[a,b]$, 
 then the area of the region bounded by the graphs of $f,g,x=a,x=b$ given by
 
 $A=\int _{a}^{b}[f(x)-g(x)]dx$

Exercises

1) Find the area of the region bounded by the graph of $f(x)=x^{2}$ and the graph of $g(x)=x^{3}$.

Solution:
Find the bound of the region by setting $f(x)=g(x)$, so $x^{2}=x^{3}$, hence $x^{3}-x^{2}=0$, then $x^{2}(x-1)=0$, therefore $x=0$ and $x=1$
Check which function is the top function by choosing one number in between the bound and plug in:
Pick $x={\frac {1}{2}}$, so $f({\frac {1}{2}})={\frac {1}{4}}$, and $g({\frac {1}{2}})={\frac {1}{8}}$. Therefore, $f(x)$ will be the top function.
Failed to parse (syntax error): {\displaystyle \text{Area}=\int_0^1 [f(x)-g(x)]dx=\int_0^1 [x^2-x^3]dx=[\frac{1}{3} x^3-\frac{1}{4} x^4]\biggr_0^1=\frac{1}{3} (1)^3-\frac{1}{4} (1)^4=\frac{1}{12}}

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