Example 1
Find the area bounded by the curve y = 9 - x² and the x-axis.
 

There are two methods for finding the area bounded by curves in rectangular coordinates. These are...

1. by using a horizontal element (called strip) of area, and
2. by using a vertical strip of area.

The strip is in the form of a rectangle with area equal to length × width, with width equal to the differential element. To find the total area enclosed by specified curves, it is necessary to sum up a series of rectangles defined by the strip.
 

Using Horizontal Strip

Partial Fractions

Functions of $x$ that can be expressed in the form $P(x)/Q(x)$, where both $P(x)$ and $Q(x)$ are polynomials of $x$, is known as rational fraction. A rational fraction is known to be a proper fraction if $P(x)$ is less in degree power than $Q(x)$. Example of proper fraction is...
 

Trigonometric substitution is employed to integrate expressions involving functions of (a² − u²), (a² + u²), and (u² − a²) where "a" is a constant and "u" is any algebraic function. Substitutions convert the respective functions to expressions in terms of trigonometric functions. The substitution is more useful but not limited to functions involving radicals.
 

Problem
Evaluate $\displaystyle \int \dfrac{(8x + 1) \, dx}{\sqrt{4x - 3}}$
 

In algebraic substitution we replace the variable of integration by a function of a new variable. A change in the variable on integration often reduces an integrand to an easier integrable form.
 

$$ \int f(g(x)) \, g'(x) \, dx = \int f(u) \, du $$

where $u = g(x)$
 

When $u$ and $v$ are differentiable functions of $x$, $d(uv) = u \, dv + v \, du\,$ or $\,u \, dv = d(uv) - v \, du$. When this is integrated we have
 

Integration by Parts
Integration by Substitution
Integration of Rational Fractions
Change of Limits with Change of Variable
 

In applying the formula (Example: Formula 1 below), it is important to note that the numerator du is the differential of the variable quantity u which appears squared inside the square root symbol. We mentally put the quantity under the radical into the form of the square of the constant minus the square of the variable.
 
1. $\displaystyle \int \dfrac{du}{\sqrt{a^2 - u^2}} = \arcsin \, \dfrac{u}{a} + C, \,\,\, a > 0$

2. $\displaystyle \int \dfrac{du}{a^2 + u^2} = \dfrac{1}{a}\arctan \, \dfrac{u}{a} + C$

3. $\displaystyle \int \dfrac{du}{u\sqrt{u^2 - a^2}} = \dfrac{1}{a} {\rm arcsec} \, \dfrac{u}{a} + C$
 

Basic Formulas

1. $\displaystyle \int \sin u \, du = -\cos u + C$

2. $\displaystyle \int \cos u \, du = \sin u + C$

3. $\displaystyle \int \sec^2 u \, du = \tan u + C$

4. $\displaystyle \int \csc^2 u \, du = -\cot u + C$

5. $\displaystyle \int \sec u \, \tan u \, du = \sec u + C$

6. $\displaystyle \int \csc u \, \cot u \, du = -\csc u + C$