Example 3
Find the area bounded by the curve x = y² + 2y and the line x = 3.
 

Example 2
Find the area bounded by the curve a² y = x³, the x-axis and the line x = 2a.
 

Solids of Revolution by Integration

The solid generated by rotating a plane area about an axis in its plane is called a solid of revolution. The volume of a solid of revolution may be found by the following procedures:
 

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

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 $u = g(x)$