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

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

Areas and Volumes 277

Or, find the area of each rectangle:

Area of RectI=(f(0))Δx 1 =(1)

(

1

2

)

=

1

2

.

Area of RectII= f(0.5)Δx 2 =((.5)^2 +1)

(

1

2

)

= 0. 625.

Area of RectIII= f(1)Δx 3 =(1^2 +1)

(

1

2

)

= 1.

Area of RectIV= f(1.5)Δx 4 =(1. 52 +1)

(

1

2

)

= 1. 625.

Area of (RectI+RectII+RectIII+RectIV)= 3. 75.

Thus, the approximate area under the curve off(x) is 3.75.

Example 2

Find the approximate area under the curve off(x)=

√

xfromx=4tox=9 using 5 right-
endpoint rectangles. (See Figure 13.2-3.)

0

IIIIII IVV

x

y f(x) = x

456789

Figure 13.2-3

LetΔxibe the length of theith rectangle. The lengthΔxi=

9 − 4

5

=1;xi= 4 +(1)i=

4 +i.

Area of RectI= f(x 1 )Δx 1 = f(5)(1)=

√

5.

Area of RectII= f(x 2 )Δx 2 = f(6)(1)=

√

6.

Area of RectIII= f(x 3 )Δx 3 = f(7)(1)=

√

7.

Area of RectIV= f(x 4 )Δx 4 = f(8)(1)=

√

8.

Area of Rectv= f(x 5 )Δx 5 = f(9)(1)=

√

9 = 3.

∑^5

i= 1

(Area of RectI)=

√

5 +

√

6 +

√

7 +

√

8 + 3 = 13. 160.