Math 22 Area and the Fundamental Theorem of Calculus

Definition of a Definite Integral

 Let $f$ be nonnegative and continuous on the closed interval $[a,b]$. 
 The area of the region bounded by the graph of $f$, the x-axis, 
 and the lines $x=a$ and $x=b$ is denoted by
 ${\text{Area}}=\int _{a}^{b}f(x)dx$
 
 The expression $\int _{a}^{b}f(x)dx$ is called the definite integral from a to b, 
 where a is the lower limit of integration and b is the upper limit of integration.

The Fundamental Theorem of Calculus

 If $f$ is nonnegative and continuous on the closed interval [a,b], then
 
 $\int _{a}^{b}f(x)dx=F(b)-F(a)$
 
 where $F(x)$ is any function such that $F'(x)=f(x)$ for all $x$ in [a,b]

Notation $\int _{a}^{b}f(x)dx=F(x){\Biggr |}_{a}^{b}=F(b)-F(a)$

Properties of Definite Integrals

 Let $f$ and g be continuous on the closed interval [a,b].

 1.$\int _{a}^{b}kf(x)dx=f\int _{a}^{b}f(x)dx$ for $k$ is constant.
 
 2.$\int _{a}^{b}[f(x)\pm g(x)]dx=\int _{a}^{b}f(x)dx\pm \int _{a}^{b}g(x)dx$
 
 3.$\int _{a}^{b}f(x)d=\int _{a}^{c}f(x)dx+\int _{c}^{b}f(x)dx$ for $a<c<b$
 
 4.$\int _{a}^{a}f(x)dx=0$
 
 5.$\int _{a}^{b}f(x)dx=-\int _{b}^{a}f(x)dx$

Exercises Find the area of the region bounded by the x-axis and the graph of

1) $y=5x$ when $0\leq x\leq 3$

Solution:
$\int _{0}^{3}5xdx=3\int e^{x}={\frac {5}{2}}x^{2}{\Biggr |}_{0}^{3}=[{\frac {5}{2}}(3)^{2}-{\frac {5}{2}}(0)^{2}]={\frac {45}{2}}$

2) $\int 3e^{3x}dx$

Solution:
Let $u=3x$, so $du=3dx$, so $dx={\frac {du}{3}}$
Consider $\int 3e^{3x}dx=\int 3e^{u}{\frac {du}{3}}=\int e^{u}du=e^{u}+C=e^{3x}+C$

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