One third of a second after the ball leaves the pitcher’s hand, the distance it travels is equal to
This value is just over three quarters of the way to home plate. The speed of the ball is
This speed translates to approximately 95 mph—a major-league fastball.
Surface area generated by a parametric curve
Recall the problem of finding the surface area of a volume of revolution. In
Curve Length and Surface Area , we derived a formula for finding the surface area of a volume generated by a function
from
to
revolved around the
x -axis:
We now consider a volume of revolution generated by revolving a parametrically defined curve
around the
x -axis as shown in the following figure.
The analogous formula for a parametrically defined curve is
provided that
is not negative on
Finding surface area
Find the surface area of a sphere of radius
r centered at the origin.
We start with the curve defined by the equations
This generates an upper semicircle of radius
r centered at the origin as shown in the following graph.
When this curve is revolved around the
x -axis, it generates a sphere of radius
r . To calculate the surface area of the sphere, we use
[link] :
This is, in fact, the formula for the surface area of a sphere.
The derivative of the parametrically defined curve
and
can be calculated using the formula
Using the derivative, we can find the equation of a tangent line to a parametric curve.
The area between a parametric curve and the
x -axis can be determined by using the formula
The arc length of a parametric curve can be calculated by using the formula
The surface area of a volume of revolution revolved around the
x -axis is given by
If the curve is revolved around the
y -axis, then the formula is
Key equations
Derivative of parametric equations
Second-order derivative of parametric equations
Area under a parametric curve
Arc length of a parametric curve
Surface area generated by a parametric curve
For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line.
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
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