Review 2: Definite versus Indefinite Integral

Indefinite Integral	Definite Integral
$\int f (x) d x$ $\int f(x)dx$ is a function of $x$ $x$ (actually it is a family of functions)	$\int_{a}^{b} f (x) d x$ $\int_a^b f(x)dx$ is a number
By definition $\int f (x) d x$ $\int f(x)dx$ is any function $F (x)$ $F(x)$ who derivative is $f (x)$ $f(x)$	$\int_{a}^{b} f (x) d x$ $\int_a^b f(x)dx$ was defined in terms of Riemann sums and can be interpreted as "area under the graph of $y = f (x)$ $y=f(x)$ when $f (x) \geq 0.$ $f(x)\ge 0.$
If $F (x)$ $F(x)$ is an antiderivative of $f (x)$ $f(x)$ , then so is $F (x) + C$ $F(x) + C$ . Therefore $\int f (x) d x = F (x) + C$ $\int f(x)dx=F(x)+C$ , an indefinite integral contains a constant ("+C").	$\int_{a}^{b} f (x) d x$ $\int_a^b f(x)dx$ is one uniquely defined number; a definite integral does not contain an arbitrary constant.

6.1 Examples

a) $F (t) = \int 3 t^{2} d t = t^{3} + C .$ $F(t) = \int 3t^2\; dt=t^3+C.$

Here, when we write and compute $\int 3 t^{2} d t$ $\int 3t^2\; dt$ we are using the letter or variable $t$ $t$ as an object or place holder that helps with our computation while we are also using it as a variable to represent the input of the function $F (t)$ $F(t)$ .

b) $LaTeX: \frac{dF(t)}{dt}=\frac{d}{dt} \left(\int 3t^2\; dt\right)=3t^2$ $\frac{dF(t)}{dt}=\frac{d}{dt} \left(\int 3t^2\; dt\right)=3t^2$

a) $LaTeX: \int_1^2 3t^2 \; dt= t^3 \Big |_1^2= 2^3-1^3=7$ $\int_1^2 3t^2 \; dt= t^3 \Big |_1^2= 2^3-1^3=7$

b) $LaTeX: \frac{d}{dt} \left(\int_1^2 3t^2 \; dt\right)=\frac{d}{dt}(7)=0$ $\frac{d}{dt} \left(\int_1^2 3t^2 \; dt\right)=\frac{d}{dt}(7)=0$

In both cases here, letter t is being used as placeholder (dummy variable) for computing an antiderivative which is used in the calculations.

3. $LaTeX: \int_1^x 3t^2 \; dt= t^3 \Big |_1^x= x^3-1^3$ $\int_1^x 3t^2 \; dt= t^3 \Big |_1^x= x^3-1^3$

Letter t is a dummy variable and x is a variable.