Approximating Integrals on the TIs

1  Set up the approximating sums

• Define dd(n)=(b-a)/n

• Define xx(k,n)=a+k*dd(n)

• Define lrr(n)=dd(n)*å(f(xx(k,n)),k,0,n-1)

• Define rrr(n)=dd(n)*å(f(xx(k,n)),k,1,n)

• Define trap(n)=(lrr(n)+rrr(n))/2

• Define mx(k,n)=(xx(k-1,n)+xx(k,n))/2

• Define mi(n)=dd(n)*å(f(mx(k,n)),k,1,n)

• Define simp(n)=(trap(n/2)+2*mi(n/2))/3

2  Enter the interval and function

• Define f(x) = ....

• left endpoint STO a

• right endpoint STO b

3  Calculate the approximating sums

Entering lrr(n), rrr(n), trap(n), mi(n) and simp(n) for a specific power of n yields the approximating sums. For example, the approximating sums for the integral

ó õ 2 1  dx x

for n=5 are

• lrr(5)=.745635

• rrr(5)=.645635

• trap(5)=.695635

• mi(5)=.691908

• simp(10)=.693150

Note that the Simpson's Rule approximation is for the same integral with n=10.

4  Error Bounds

• For the Trapezoidal Rule:
  

  |ET| £ K(b-a)3 12n2 ,

  where K is an upper-bound for |f^¢¢(x)| on [a,b].

• For the Midpoint Rule:
  

  |EM| £ K(b-a)3 24n2 ,

  where K is an upper-bound for |f^¢¢(x)| on [a,b].

• For Simpson's Rule:
  

  |ES| £ K(b-a)5 180n4 ,

  where K is an upper-bound for |f⁽⁴⁾(x)| on [a,b].

Upper-bounds might be determined algebraically, estimated graphically, or derived from max/min considerations.