Approximating Integrals on the TIs

Set up the approximating sums

• Define dd(n)=(b-a)/n
• Define xx(k,n)=a+k*dd(n)
• Define lrr(n)=dd(n)* (f(xx(k,n)),k,0,n-1)
• Define rrr(n)=dd(n)* (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)* (f(mx(k,n)),k,1,n)
• Define simp(n)=(trap(n/2)+2*mi(n/2))/3

Enter the interval and function

• Define f(x) = .... Example: Define f(x)=1/x
• left endpoint STO a Store the left endpoint as a
• right endpoint STO b Store the right endpoint as b

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

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.

Error Bounds

• For the Trapezoidal Rule:

  

  where K is an upper-bound for on [a,b].

• For the Midpoint Rule:

  

  where K is an upper-bound for on [a,b].

• For Simpson's Rule:

  

  where K is an upper-bound for on [a,b].

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