Unformatted text preview: MTH405 Wk 10 Lect 2 Numerical Integration
Numerical Integration is a also another way of nding area under the
graph, but it is very grounded. Most of the time, Numerical Integration is used to nd area for functions which can not be integrated
(Yes, there are functions for which the techniques of integration we
have learnt can not be applied). Most importantly, notice that when we started Integration
topic, we stated with 100% condence that the Area under
the Graph is given by
ˆ b f (x)dx. A=
a But we did not show how this is true. (We are still trying to
answer the AREA PROBLEM)
This is true because the actual formula for nding Area under the
Graph is given by
A = lim n→+∞ n
X f (x∗k )∆x. k=1 This is called the Riemann Sum. And there is a theorem that
states that
ˆ
n
b f (x)dx = lim
a n→+∞ 1 X
k=1 f (x∗k ) · ∆x. Summary on Riemann Sum
We need to nd the area under the curve along the interval [a, b].
We use Archimedes method of exhaustion by dividing the interval
into n equal subintervals as shown below. This method is known as the Rectangle method as we have
many slim rectangles in each subinterval. Then the width of each
subinterval is given by
∆x = b−a
.
n Hence, the endpoints of the subintervals are listed as
a = x0 , x1 , x2 , . . . , xk , . . . , xn = b. To give us the the height of the rectangles, we take a point x
from each subinterval:
x∗1 , x∗2 , . . . , x∗k , . . . , x∗n−1 , x∗n , Hence the heights of the rectangles are
f (x∗1 ), f (x∗2 ), . . . , f (x∗k ), . . . , f (x∗n−1 ), f (x∗n ).
2 The area for each rectangle is given by
Ak = H × W
= f (x∗k ) · ∆x. Then taking the sum of all rectangles gives the area
A≈ n
X f (x∗k ) · ∆x. k=1 This is the APPROXIMATION FORMULA. 3 But that is just an approximation; to get a more accurate value
of area, we increase the number of subintervals n → +∞ (the more
the subintervals n, the smaller the rectangles and lesser the error).
Since we can't just say n = +∞, then we just take it to the limit:
A = lim n→∞ n
X f (x∗k ) · ∆x. k=1 This is the rigorous AREA FORMULA for area under the graph,
but this won't be covered in this course. Midpoint Approximation
Note the approximation formula given before
A≈ n
X f (x∗k ) · ∆x. k=1 This is the approximation we will be using for Midpoint, Trapezoidal
and Simpson approximations.
For Midpoint approximation, we have the following formulae to
use:
width of the subintervals, ∆x = b−a
n
1
2 choice of midpoints of the subintervals, x∗k = a + (k − )∆x
Area function, A = ˆ b
f (x)dx ≈ a Example. Evaluate Example. Evaluate ´2 1
dx
1 x ´3
1 b−a
n [f (x∗1 ) + f (x∗2 ) + . . . + f (x∗n )] where n = 4 using Midpoint Rule. (x3 + 1)dx where n = 4 using Midpoint Rule. 4...
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 Spring '16
 Alveen Chand
 Riemann sum

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