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
* Œ [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