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.
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Area by Integration
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.
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Example 3 | Plane Areas in Rectangular Coordinates
Example 3
Find the area bounded by the curve x = y2 + 2y and the line x = 3.
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Example 2 | Plane Areas in Rectangular Coordinates
Example 2
Find the area bounded by the curve a2 y = x3, the x-axis and the line x = 2a.
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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...
- by using a horizontal element (called strip) of area, and
- 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.