Averaging right and left rectangles gives us trapezoids:
(still too high)
Trapezoidal Rule: h = width of subinterval = (b – a)/n This gives us a better approximation than either left or right rectangles.
Trapezoidal Rule: h = width of subinterval = (b – a)/n To see if the Trapezoidal Rule is an overestimate, underestimate, or exact, use the Concavity Test. If f’’(x) = 0, approximation is exact. If f’’(x) > 0, approximation is an overestimate If f’’(x) < 0, approximation is an underestimate.
Example 1 Use the Trapezoidal Rule with n = 4 to estimate We must partition [1, 2] into four subintervals of equal length. x f(x)125/1636/1649/164
Compare this with the Midpoint Rule: Approximate area: (too low)0.625% error The midpoint rule gives a closer approximation than the trapezoidal rule, but in the opposite direction.
Midpoint Rule: (too low)0.625% error Trapezoidal Rule: 1.25% error (too high) Notice that the trapezoidal rule gives us an answer that has twice as much error as the midpoint rule, but in the opposite direction. If we use a weighted average: This is the exact answer!
This weighted approximation gives us a closer approximation than the midpoint or trapezoidal rules. Midpoint: Trapezoidal: twice midpointtrapezoidal
Simpson’s Rule: ( h = width of subinterval, n must be even ) Example:
Simpson’s rule can also be interpreted as fitting parabolas to sections of the curve, which is why this example came out exactly. Simpson’s rule will usually give a very good approximation with relatively few subintervals. It is especially useful when we have no equation and the data points are determined experimentally.