Page 87 - Calculus for the Clueless, Calc II

Note 3

This procedure, as well as the other two approximations, are suitable for many academic levels. Most
appropriately, they occur in either Calc I or II.

The Trapezoidal Method

This method, the only one of the three that does not actually require calculus, approximates the area under the
curve by trapezoids by approximating the top of the region by a line. Divide the region into n equal parts. See
the trapezoids. They are standing on their heights. The area of a trapezoid is ½h(b l + b 2). All of the h's are the

same. For the first trapezoid, b 1 = y 1

and b 2 = y 2; A = ½h(y 1 + y 2). For the second trapezoid, b 1 = y 2 and b 2 = y 3; A = ½h(y 2 + y 3). Notice the lower
base of the first trapezoid is the upper base of the second trapezoid. Every base is doubled except the first upper
base and the last lower base. The formula is A = ½h(y 0 + 2y 1 + 2y 2 + ... + 2y n-1 + y n).

Example 2—

Approximate dx using six equal subdivisions.

The interval is of length 4 - 1 = 3. Six equal parts? Each h = 3/6 = ½.

Parabolic Method