Integral calculus
     Applications of the definite integral
      The surface area of a solid of revolution
         The lateral surface area of a cone
         The surface area of a spherical cap
         The surface area of an ellipsoid
The surface area of a solid of revolution
The surface area generated by the segment of a curve y = f (x) between x = a and y = b rotating around the x-axis, is shown in the left figure below.
Since the infinitesimal surface area of an element of the integration,
where  y is the radius and ds is the arc length of the element of the curve, then 
   
The surface area generated by the segment of a curve x = g (y) between y = c and y = d rotating around the y-axis, is shown in the right figure above.
The surface area generated by rotating a parametric curve about the x-axis,
The lateral surface area of a cone
Example:  Find the lateral surface area of a right circular cone generated by the line (segment) through the origin and the point (h, r), where h denotes the height of the cone and r is the radius of its base, revolving around the x-axis, as shows the below figure.
Solution:  The equation of the generating line
then,
 
The surface area of a spherical cap
Example:  Find the surface area of a spherical cap, with the height h, generated by the portion of the right semicircle rotating around the y-axis, as is shown in the below figure.
Solution:  The equation of the right semicircle
then,
 
The surface area of an ellipsoid
Example:  Find the surface area of an ellipsoid generated by the ellipse b2x2 + a2y2 = a2b2 rotating around the x-axis, as shows the below figure.
Solution:  The equation of the upper half of the ellipse and its derivative
the arc length of the element of the curve
where,  e and e denote the linear and the numerical eccentricity respectively. Therefore,
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