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If f(x) 0 for a x b, the Riemann sum represents an approximation of the area under the graph of f(x), above the x-axis, from x = a to x = b. The diagram shows the Riemann sum of the function f(x) = x2 over the interval [1, 2] as the area enclosed by four approximating rectangles of equal width.
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The Riemann sum, represented by the gray area enclosed by the rectangles, offers only an approximation to the area under the curve. However, as the width of each rectangle shrinks, the approximation gets better and the exact area under the curve is approached as a limit. The definite integral of f(x) over [a, b] is defined in many calculus texts by
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f ( x )dx = lim
||P|| 0
f ( x * ) x
i i =1
where
|| P || = max xi
1 i n
The condition || P || 0 guarantees that the lengths of all subintervals shrink toward 0 as we take more and more subintervals. If all subintervals are of equal length, this condition is equivalent to n . For convenience we shall only consider subintervals of equal length. However, in theory, this need not be the case.
EXAMPLE 16 We will consider the function f (x) = sin x on the interval [0, /2]. Because
good example for comparative purposes. (a) We use 100 subintervals and choose xi* to be the left endpoint of each subinterval. f[x_] = Sin[x]; a = 0; b = /2; n = 100; x = (b a)/n;
/2
sin x dx = 1, this is a
Since each x has the same value, subscripts are not necessary.
xstar[i_] = a + (i 1) x;
f[xstar[i]] x //N
0.992125 (b) We choose the value of xi* to be the right endpoint of each subinterval. (This time we expect an overapproximation.) f[x_] = Sin[x]; a = 0; b = /2; n = 100; x = (b a)/n; xstar[i_]= a + i x;
f[xstar[i]] x //N
To improve the accuracy of the approximation offered in Example 16, we can choose the value of xi* to be the midpoint of each subinterval. This leads to an approximation method called the midpoint rule.
EXAMPLE 17
f[x_] = Sin[x] ; a = 0; b = o/2; n = 100; x = (b a)/n;. xstar[i_] = a +(i .5) x;
f[xstar[i]] x
/ /N
1.00001 As expected, the accuracy of the approximation improves.
Another simple approximation method, called the trapezoidal rule, improves accuracy by connecting the points on the curve corresponding to the points of subdivision with line segments, forming trapezoidal approximations of the area in place of rectangular approximations.
Integral Calculus
f (x) 1 0.8 0.6 0.4 0.2 x
/2
Trapezoidal approximation to
sin x dx using four trapezoids.
The area enclosed by a trapezoid with base x and sides A and B is x ( A + B). 2 Thus, the area enclosed by the trapezoid constructed in the ith interval, xi [xi 1, xi], is f ( xi 1 ) + f ( xi ) . 2 The total trapezoidal area, obtained by adding the individual areas, is
f (xi 1) f (xi)
xi 1
x x x1 x [ f ( x 0 ) + f ( x1 )] + 2 [ f ( x1 ) + f ( x 2 )] + 3 [ f ( x 2 ) + f ( x3 )] + . . . + n [ f ( x n 1 ) + f ( x n )] 2 2 2 2 If all intervals have the same length, x, this reduces to x [ f ( x 0 ) + f ( x1 )] + [ f ( x1 ) + f ( x 2 )] + [ f ( x 2 ) + f ( x3 )] + . . . + [ f ( x n 1 ) + f ( x n )] 2 or x f ( x 0 ) + 2 f ( x1 ) + 2 f ( x 2 ) + 2 f ( x3 ) + . . . + 2 f ( x n 1 ) + f ( x n ) 2
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