Graphics Programs Reference
In-Depth Information
8. Evaluate
1
sin
x
√
x
dx
0
with Romberg integration.
Hint
:use transformation of variable to eliminate the
indeterminacyat
x
=
0.
9. Show that if
y
=
f
(
x
) is approximatedbyanaturalcubicspline with evenly spaced
knots at
x
1
,
x
2
,...,
x
n
, the quadratureformulabecomes
h
2
(
y
1
+
I
=
2
y
2
+
2
y
3
+···+
2
y
n
−
1
+
y
n
)
h
3
24
−
(
k
1
+
2
k
2
+
k
3
+···+
2
k
n
−
1
+
k
n
)
where
h
is the spacing of the knots and
k
y
. Note that the first part is the
composite trapezoidal rule; the second part may be viewedas a“correction” for
curvature.
=
10.
Use a computerprogram to evaluate
π/
4
dx
√
sin
x
with Romberg integration.
Hint
:use the transformation sin
x
0
t
2
.
=
4
√
L
11.
The period of a simple pendulum of length
L
is
τ
=
/
gh
(
θ
0
), where
g
is
the gravitational acceleration,
θ
0
represents the angular amplitude and
π/
2
d
θ
θ
0
)
=
1
h
(
sin
2
(
2)sin
2
0
−
θ
0
/
θ
Compute
h
(15
◦
),
h
(30
◦
) and
h
(45
◦
), and compare these values with
h
(0)
=
π/
2 (the
approximationused for small amplitudes).
12.
r
q
a
P
The figure shows an elastichalf-space thatcarries uniform loading of intensity
q
overacircular area of radius
a
. The vertical displacementofthesurface at point
P
can be shown to be
w
0
π/
2
0
cos
2
θ
w
(
r
)
=
(
r
d
θ
r
≥
a
sin
2
/
a
)
2
−
θ
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