Area by Integration

Example 4 | Plane Areas in Rectangular Coordinates

Example 4
Solve the area bounded by the curve y = 4x - x2 and the lines x = -2 and y = 4.
 

02 Area Bounded by the Lemniscate of Bernoulli r^2 = a^2 cos 2θ

Example 2
Find the area bounded by the lemniscate of Bernoulli r2 = a2 cos 2θ.
 

01 Area Enclosed by r = 2a sin^2 θ

Example 1
Find the area enclosed by r = 2a sin2 θ.
 

Example 3 | Plane Areas in Rectangular Coordinates

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

Example 2 | Plane Areas in Rectangular Coordinates

Example 2
Find the area bounded by the curve a2y = x3, the x-axis and the line x = 2a.
 

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...
 

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

 

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

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