Part II: The Answers
592
Answers
901–1,001
997.
0.216
You want to approximate
sin x dx
0
1 2
∫
with n = 4. Using the trapezoid rule, you have
∆ x =
− = =
1
2
0
4
1
8
0 125
.
so that
sin
.
( )
( .
)
( . )
( .
) ( . )
x dx
f
f
f
f
f
0
1 2
0 125
2
0 2 0 125 2 0 25 2 0 375
0 5
∫
≈
+
+
+
+
(
) )
=
+
+
+
+
)
(
≈
0 125
2
0 2
0 125 2
025 2
0375
05
0 216
.
sin
s in .
s in .
sin .
sin .
.
998.
18.817
Recall that Simpson’s rule states
f x
x f x
f x
f x
f x
f x
f x
f
n
n
( ) ≈
( )+ ( )+ ( )+ ( )+ + (
)+ ( )+
−
−
∆
3
4
2
4
2
4
0
1
2
3
2
1
...
x x n
a
b
( ))
(
∫
where n is even, ∆ x b a
n
= − , and x i = a + iΔx.
You want to approximate
1
3
3
0
6
+
∫
x dx with n = 6. Using Simpson’s rule with n = 6 gives
you ∆ x b a
n
= − = − =
6 0
6
1 so that
1
1
3
0 4 1 2 2 4 3 2 4 4 5
6
1
3
1
3
3
0
6
+
≈
+
+
+
+
+
+
)
(
=
∫
x dx
f
f
f
f
f
f
f
( )
( )
( )
( )
( )
( ) ( )
+ + +
+ +
+ +
+
+
+
+
+ + + )
(
≈
0
4 1 1 2 1 2
4 1 3
2 1 4
4 1 5
1 6
18 817
3
3
3
3
3
3
3
3
3
3
3
3
3
3
.
999.
0.107
You want to approximate
ln x
x
dx
1
2
1
2
+
∫
with n = 4. Using Simpson’s rule with n = 4 gives
you ∆ x = − = =
2 1
4
1
4
0 25
. so that
ln
.
( )
( . )
( . )
( . ) ( )
.
x
x
dx
f
f
f
f
f
1
0 25
3
1 4 1 25 2 1 5 4 1 75
2
0
2
1
2
+
≈
+
+
+
+
)
(
=
∫
2 25
3
1
1 1
4
1 25
1 1 25
2
1 5
1 1 5
4
1 75
1
2
2
2
ln
ln( . )
( . )
ln( . )
( . )
ln( . )
+
+
+
+ +
+
+ +
+ +

 

 
≈
( . )
ln
.
1 75
2
1 2
0 107
2
2