Example 5 | Plane Areas in Rectangular Coordinates
Example 5
Find the area between the curves 2x2 + 4x + y = 0 and y = 2x.
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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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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.
Example 2 | Volumes of Solids of Revolution
Example 2
Find the volume generated when the area in Example 1 will revolve about the y-axis.
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Example 1 | Volumes of Solids of Revolution
Example 1
Find the volume of the solid generated when the area bounded by the curve y2 = x, the x-axis and the line x = 2 is revolved about the x-axis.
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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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Volumes of Solids of Revolution | Applications of Integration
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:
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...
$\displaystyle A = \frac{1}{2}{\int_{\theta_1}^{\theta_2}} r^2 \, d\theta$
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