
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
xaxis,
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
yaxis,
is shown in the right figure above.

The surface area generated by rotating
a parametric curve about the
xaxis,



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
xaxis,
as shows the below figure.

Solution: The equation of the
generating line



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 yaxis,
as
is shown in the below figure.

Solution: The equation of the right semicircle



The surface area of an
ellipsoid

Example: Find the surface area of an
ellipsoid generated by the ellipse b^{2}x^{2}
+ a^{2}y^{2 }= a^{2}b^{2}
rotating around
the
xaxis,
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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