Arc Length of Smooth, Planar Curve
AP Calculus BC · Flashcards · Application
AP CALCULUS BC · FLASHCARDS
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QUESTION
Determine the arc length of
r
=
1
+
sin
(
θ
)
r = 1 + \sin(\theta)
r
=
1
+
sin
(
θ
)
from
θ
=
0
\theta = 0
θ
=
0
to
θ
=
π
\theta = \pi
θ
=
π
.
QUESTION
Find the total distance traveled by a particle with velocity
v
(
t
)
=
3
t
v(t) = 3t
v
(
t
)
=
3
t
from
t
=
0
t = 0
t
=
0
to
t
=
2
t = 2
t
=
2
.
QUESTION
What is the curvature of the parametric curve
x
(
t
)
=
t
,
y
(
t
)
=
t
2
x(t) = t, y(t) = t^2
x
(
t
)
=
t
,
y
(
t
)
=
t
2
?
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ANSWER
κ
=
2
(
1
+
4
t
2
)
3
/
2
\kappa = \frac{2}{(1 + 4t^2)^{3/2}}
κ
=
(
1
+
4
t
2
)
3/2
2
. Applies the parametric curvature formula with
x
′
(
t
)
=
1
,
y
′
(
t
)
=
2
t
,
y
′
′
(
t
)
=
2
x'(t) = 1, y'(t) = 2t, y''(t) = 2
x
′
(
t
)
=
1
,
y
′
(
t
)
=
2
t
,
y
′′
(
t
)
=
2
.
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