- The goal of Numerical Integral Calculus is to compute the integral of a function.
Riemann Sum
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One approximation can be obtained using the Riemann Integral. Let
be a function. Also let be a partition over this integral such that Each sample point
can be chosen from anywhere in the interval Also we define - Note that if the intervals are spaced out, we have that
- Note that if the intervals are spaced out, we have that
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More sample points chosen tends to give better approximations.
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The Left Rule involves choosing
Then
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The Right Rule involves choosing
Then
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Both the left and right Riemann sum have an error bound that is
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Proof: Let
and be the left and right sums respectively. Observe that Also note that this is an upper bound for both the left and the right sums. In particular, one sum will overestimate and one will underestimate so that if the true integral is
, we have without loss of generality So that both
and . Therefore the error for both is
. -
Proof: An alternate proof obtains the precise bound. Let
be the maximum of the first derivative in the interval. Assume is continuous and well defined over the interval. We proceed for the left Riemann sum but the proof is the same for the right.
\sum_{i=1}^n \frac{f’(\xi_i) h^2}{2}=\frac{h(b-a)}{2}\left(\frac{1}{n}\sum_{i=1}^n f’(\xi_i)\right) \le \frac{Mh(b-a)}{2}
-
-
The Midpoint Rule involves choosing
Then
- Suppose
is continuous over and the second derivative is with maximum absolute value . Then the midpoint rule has an error bound that is . -
Proof: To be more precise, the error
is bounded by Let
. If we start with a single interval and use the Taylor Series expansion In the above,
is the point such that by the Mean Value Theorem, is the average of all in the interval. The proof follows by taking the sum of the above over all intervals. The bound is obtained by considering the maximum absolute value of
over the interval. In particular
-
- Suppose
-
The Lower rule involves choosing
Thus
-
The Upper rule involves choosing
Thus
Curve Approximation
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The Trapezoid Rule approximates the Riemann sum using trapezoidal cuts. In particular, we make use of secant lines to approximate the shape of the curve.
Given interval
, the area of the trapezoid in the region is defined as And the integral is approximated as
- Suppose
is continuous over and the second derivative is with maximum absolute value . Then the trapezoid rule has an error bound that is . - Proof The proof proceeds similarly to the proof in the Midpoint Rule. We can use the same proof sketch to show that the Midpoint rule has error bound
Where
is the maximum absolute value of .
- Proof The proof proceeds similarly to the proof in the Midpoint Rule. We can use the same proof sketch to show that the Midpoint rule has error bound
- Suppose
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Simpson’s Rule uses quadratic approximations of the curve to obtain the integral. It is like the Trapezoidal rule but using tangent quadratic curves rather than tangent lines. It also uses three points for the approximation
Given interval
, the area of the trapezoid in the region is defined as The integral is then approximated as
- Simpson’s Rule gives a fourth order approximation. That is, the error bound is
. - Proof: We prove a related result. Simpson’s rule is found by taking the weighted average of the Midpoint Rule approximation and the Trapezoid Rule Approximation. Denote these by
and respectively. Then the Simpson Rule approximation is given by Expanding this weighted average gives us the desired fourth order approximation term.
- Proof: We prove a related result. Simpson’s rule is found by taking the weighted average of the Midpoint Rule approximation and the Trapezoid Rule Approximation. Denote these by
- Simpson’s Rule gives a fourth order approximation. That is, the error bound is