As illustrated in the accompanying figure, a sinusoidal cut is made in the top of a cylindrical tin can. Suppose that the base is modelled by the parametric equations x = cos t , y = sin t , z = 0 ( 0 ≤ t ≤ 2 π ) , and the height of the cut as a functions of t , z = 0 ( 0 ≤ t ≤ 2 π ) , and the height of the cut as a functions of t is z = 2 + 0.5 sin 3 t . (a) Use a geometric argument to find the lateral surface area or the cut can. (b) Write down a line integral for the surface area. (c) Use the line integral to calculate the surface area.
As illustrated in the accompanying figure, a sinusoidal cut is made in the top of a cylindrical tin can. Suppose that the base is modelled by the parametric equations x = cos t , y = sin t , z = 0 ( 0 ≤ t ≤ 2 π ) , and the height of the cut as a functions of t , z = 0 ( 0 ≤ t ≤ 2 π ) , and the height of the cut as a functions of t is z = 2 + 0.5 sin 3 t . (a) Use a geometric argument to find the lateral surface area or the cut can. (b) Write down a line integral for the surface area. (c) Use the line integral to calculate the surface area.
As illustrated in the accompanying figure, a sinusoidal cut is made in the top of a cylindrical tin can. Suppose that the base is modelled by the parametric equations
x
=
cos
t
,
y
=
sin
t
,
z
=
0
(
0
≤
t
≤
2
π
)
,
and the height of the cut as a functions of
t
,
z
=
0
(
0
≤
t
≤
2
π
)
,
and the height of the cut as a functions of
t
is
z
=
2
+
0.5
sin
3
t
.
(a) Use a geometric argument to find the lateral surface area or the cut can.
(b) Write down a line integral for the surface area.
(c) Use the line integral to calculate the surface area.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
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