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strip areas to obtain the value of the integral. This technique is referred to as
the midpoint rule.
We can justify the above algorithm by recalling the Mean Value Theorem of
Calculus, which states that:

∫ b fx dx() = ( b a f c) ( )− (4.4)
a

where c ∈ [a, b]. Thus, if we divide the interval of integration into narrow sub-
intervals, then the total integral can be written as the sum of the integrals over
the subintervals, and we approximate the location of c in a particular sub-
interval by the midpoint between its boundaries.

Example 4.5
Use the above algorithm to compute the value of the definite integral of the
function sin(x) from 0 to π.

Solution: Edit and execute the following program:

dx=pi/200;
x=0:dx:pi-dx;
xshift=x+dx/2;
yshift=sin(xshift);
Int=dx*sum(yshift)
You get for the above integral a result that is within 1/1000 error from the
analytical result.

In-Class Exercises

Find numerically, to a 1/10,000 accuracy, the values of the following definite
integrals:

∫ ∞ 1
Pb. 4.21 0 x + 1 dx
2

∫ ∞ 2
Pb. 4.22 exp(−x )cos( x2 )dx
0
∫ π /2 6 7
Pb. 4.23 0 sin ( )cos ( )x x dx

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