Graphics Reference
In-Depth Information
Figure 6.19.
Area subtended by curve from origin.
b
Ú
(
)
A
=
gg
¢ -
g g
1 2 .
¢
(6.32)
12
a
Proof. See Figure 6.19. Let a = t 0 , t 1 ,..., t k = b be a partition of [a,b]. Let DA i be the
signed area of the triangle 0 g(t i )g(t i+1 ). Since
(
) -
() = (
)
(
)
g
t
g
t
g
t
*
t
-
t
,
i
+
1
i
i
i
+
1
i
for some t i * Π[t i ,t i+1 ], we get that
()
¢ (
g
t
t
Ê
Á
ˆ
˜
i
= ()
(
)
DA
12
det
t
-
t
i
i
+
1
i
)
g
*
i
by Formula 6.7.3. The sign of DA i will be positive if the curve is moving in a coun-
terclockwise direction about the origin at g(t i ) and negative otherwise. Adding up all
the DA i is clearly an approximation to A and a Riemann sum that converges to the
integral in the formula as the norm of the partition goes to zero.
6.8
Circle Formulas
6.8.1 Formula. Let p 1 , p 2 , and p 3 be noncollinear points in R 3 . Let a = p 2 - p 1 and
b = p 3 - p 1 . The unique circle that contains the points p i has center o = p 1 + c , where
(
)
(
)
+ (
)
(
)
b•b a•a
-
a•b a
a•a b•b
-
a•b b
c
=
,
(6.32)
2
2
ab
¥
and radius r, where
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