5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 16:1

Areas and Volumes 279

Trapezoidal Approximations

Another method of approximating the area under a curve is to use trapezoids. (See
Figure 13.2-5.)

a = x 0 x 1 x 2 b = x 3

y

0

x

f(x)

Figure 13.2-5

Formula for Trapezoidal Approximation
Iff is continuous, the area under the curve off fromx=atox=bis:

Area≈

b−a

2 n

[

f(x 0 )+ 2 f(x 1 )+ 2 f(x 2 )+···+ 2 f(xn− 1 )+f(xn)

]

.

Example 1

Find the approximate area under the curve of f(x)=cos

(

x

2

)

fromx=0tox=π,

using 4 trapezoids. (See Figure 13.2-6.)

0

1

4

2 x

π π 2 34 π x

y

f(x) = cos

π

(

(

Figure 13.2-6

Sincen=4,Δx=
π− 0
4

=

π

4

.