Definite Integral - Calculus - Practice Questions & MCQ

Let f be a function of x defined on the closed interval [a, b] and F be another function such that $\frac{d}{d x}(F(x))=f(x)$   for all x in the domain of f, then 

$\int_a^b f(x) d x=[F(x)+c]_a^b=F(b)-F(a)$

is called the definite integral of the function f(x) over the interval [a, b],  where a is called the lower limit of the integral and b is called the upper limit of the integral. 

Geometrical Interpretation of Definite Integral 

If the function f(x) ≥ 0 for all x in [a, b],  then the definite integral $\int_a^b f(x) d x$  is equal to the area bounded by the curve f(x) at the top, the x-axis at the bottom, the line x=a to the left, and the line x=b at right.

The area above the x-axis is taken with a positive sign and the area below the x-axis is taken with a negative sign. For the figure given below,

$
\int_a^b f(x) d x=\text { Area (OLA) - Area (AQM) - Area (MRB) }+ \text { Area (BSCD) }
$

Working Rule to evaluate definite Integral $\int_a^b f(x) d x$
1. First find the indefinite integration $\int f(x) d x$ and suppose the result be $\mathrm{F}(\mathrm{x})$
2. Next find the $F(\mathrm{~b})$ and $\mathrm{F}(\mathrm{a})$
3. And, finally the value of the definite integral is obtained by subtracting $F(a)$ from $F(b)$.
Thus, $\quad \int_a^b f(x) d x=[F(x)]_a^b=F(b)-F(a)$.