Approximate Integration ~ p. 1
J. Ahrens 2001-2006
NUMERICAL INTEGRATION
Objective: Approximate integrals using trapezoidal and Simpson’s rules
Numerical integration can be useful
• If it is impossible (for us, at least) to integrate the function
• If there is no formula for the function
Previous methods of approximate integration using rectangles: left, right, and midpoint
• Left rectangles: Area under the graph of
y = f(x) from x = 1 to x = 5
using 4 subintervals:
x1
Δ
=
4
A
L [f(1) f(2) f(3) f(4)](1) 24.8≈= + + + =
• Right rectangles: Area under the graph of
y = f(x) from x = 1 to x = 5
using 4 subintervals:
x1
Δ
=
4
A
R [f(2) f(3) f(4) f(5)](1) 26.24≈= + + + =
• Midpoint rectangles: Area under the graph of
y = f(x) from x = 1 to x = 5
using 4 subintervals:
x1
Δ
=
4
AM [f
(
1.5
)
f
(
2.5
)
f
(
3.5
)
f
(
4.5
)
]
(
1
)
26.36≈= + + + =
Use left, right, and midpoint rectangles to approximate using 6
3
5
0
1dy
1y+
∫
rectangles. Round answers to 6 decimal places.
