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d d x a x f ( u ) d u = f ( x ) .

Therefore

s ( t ) = d d t [ s ( t ) ] = d d t [ 0 t 140 2 + ( −32 v + 2 ) 2 d v ] = 140 2 + ( −32 t + 2 ) 2 = 1024 t 2 128 t + 19604 = 2 256 t 2 32 t + 4901 .

One third of a second after the ball leaves the pitcher’s hand, the distance it travels is equal to

s ( 1 3 ) = ( 1 / 3 2 1 32 ) 1024 ( 1 3 ) 2 128 ( 1 3 ) + 19604 1225 4 ln | ( −32 ( 1 3 ) + 2 ) + 1024 ( 1 3 ) 2 128 ( 1 3 ) + 19604 | + 19604 32 + 1225 4 ln ( 2 + 19604 ) 46.69 feet .

This value is just over three quarters of the way to home plate. The speed of the ball is

s ( 1 3 ) = 2 256 ( 1 3 ) 2 16 ( 1 3 ) + 4901 140.34 ft/s .

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 y = f ( x ) from x = a to x = b , revolved around the x -axis:

S = 2 π a b f ( x ) 1 + ( f ( x ) ) 2 d x .

We now consider a volume of revolution generated by revolving a parametrically defined curve x = x ( t ) , y = y ( t ) , a t b around the x -axis as shown in the following figure.

A curve is drawn in the first quadrant with endpoints marked t = a and t = b. On this curve, there is a point marked (x(t), y(t)). There is a circle with an arrow drawn around the x axis that seems to indicate a rotation about the x axis, and there is a shape that accompanies that curve that seems to be what you would obtain if you rotated the curve about the x axis.
A surface of revolution generated by a parametrically defined curve.

The analogous formula for a parametrically defined curve is

S = 2 π a b y ( t ) ( x ( t ) ) 2 + ( y ( t ) ) 2 d t

provided that y ( t ) is not negative on [ a , b ] .

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

x ( t ) = r cos t , y ( t ) = r sin t , 0 t π .

This generates an upper semicircle of radius r centered at the origin as shown in the following graph.

A semicircle is drawn with radius r. On the graph there are also written three equations: x(t) = r cos(t), y(t) = r sin(t), and 0 ≤ t ≤ π.
A semicircle generated by parametric equations.

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] :

S = 2 π a b y ( t ) ( x ( t ) ) 2 + ( y ( t ) ) 2 d t = 2 π 0 π r sin t ( r sin t ) 2 + ( r cos t ) 2 d t = 2 π 0 π r sin t r 2 sin 2 t + r 2 cos 2 t d t = 2 π 0 π r sin t r 2 ( sin 2 t + cos 2 t ) d t = 2 π 0 π r 2 sin t d t = 2 π r 2 ( cos t | 0 π ) = 2 π r 2 ( cos π + cos 0 ) = 4 π r 2 .

This is, in fact, the formula for the surface area of a sphere.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the surface area generated when the plane curve defined by the equations

x ( t ) = t 3 , y ( t ) = t 2 , 0 t 1

is revolved around the x -axis.

A = π ( 494 13 + 128 ) 1215

Got questions? Get instant answers now!

Key concepts

  • The derivative of the parametrically defined curve x = x ( t ) and y = y ( t ) can be calculated using the formula d y d x = y ( t ) x ( t ) . 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 A = t 1 t 2 y ( t ) x ( t ) d t .
  • The arc length of a parametric curve can be calculated by using the formula s = t 1 t 2 ( d x d t ) 2 + ( d y d t ) 2 d t .
  • The surface area of a volume of revolution revolved around the x -axis is given by S = 2 π a b y ( t ) ( x ( t ) ) 2 + ( y ( t ) ) 2 d t . If the curve is revolved around the y -axis, then the formula is S = 2 π a b x ( t ) ( x ( t ) ) 2 + ( y ( t ) ) 2 d t .

Key equations

  • Derivative of parametric equations
    d y d x = d y / d t d x / d t = y ( t ) x ( t )
  • Second-order derivative of parametric equations
    d 2 y d x 2 = d d x ( d y d x ) = ( d / d t ) ( d y / d x ) d x / d t
  • Area under a parametric curve
    A = a b y ( t ) x ( t ) d t
  • Arc length of a parametric curve
    s = t 1 t 2 ( d x d t ) 2 + ( d y d t ) 2 d t
  • Surface area generated by a parametric curve
    S = 2 π a b y ( t ) ( x ( t ) ) 2 + ( y ( t ) ) 2 d t

For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line.

Questions & Answers

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Anassong Reply
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Bharti
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Damian Reply
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s. Reply
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s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
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of graphene you mean?
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or in general
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in general
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Graphene has a hexagonal structure
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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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