Plane Areas in Polar Coordinates | Applications of Integration

The fundamental equation for finding the area enclosed by a curve whose equation is in polar coordinates is...
 

$\displaystyle A = \frac{1}{2}{\int_{\theta_1}^{\theta_2}} r^2 \, d\theta$

 

Example 1 | Plane Areas in Rectangular Coordinates

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

Plane Areas in Rectangular Coordinates | Applications of Integration

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

Integration of Rational Fractions | Techniques of Integration

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 | Techniques of Integration

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.
 

Algebraic Substitution | Integration by Substitution

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 $u = g(x)$
 

Integration by Parts | Techniques of Integration

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
 

$\displaystyle \int u\,dv = uv - \int v\, du$

 

Chapter 3 - Techniques of Integration

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

Inverse Trigonometric Functions | Fundamental Integration Formulas

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

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