Can we generalize when Left, Right, MID and TRAP overestimate or underestimate the area under a curve? Let's look at a few pictures.
increasing function
decreasing function
SUMMARY
So far we have looked at four methods for approximating b
the definite integral :
∫ f(x) dx
a
­ Left Endpoint Approximation
­ Right Endpoint Approximation
­ Trapezoidal Rule
­ Midpoint Rule
METHOD Overestimates when
Underestimates when
Left Endpoint Approx
Right Endpoint Approx
Trapezoidal Rule
Midpoint Rule
general...it depends
And there's yet one more approximation method we'll look at: Simpson's Rule
Simpson's Rule uses parabolas to approximate the area.
As it turns out, Simpson's Rule gives a MUCH better estimate of the true area than any of the other methods we have seen so far.
Simpson's Rule
2
Recall our earlier example: ∫ (6­x2) dx = 1
LEFT(4) ≈ 4.03125
RIGHT(4) ≈ 3.28125
MID(4)≈ 3.671875
TRAP(4) ≈3.65625
Approximations:
Larger n
Left or Right
Midpoint and Trapezoid
Simpson's Rule