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

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

276 STEP 4. Review the Knowledge You Need to Score High

The area under the curve usingnrectangles of equal length is approximately:

∑n

i= 1

(area of rectangle)=

⎧

⎪⎪⎪

⎪⎪⎪

⎪⎪⎨

⎪⎪⎪

⎪⎪⎪

⎪⎪⎩

∑n

i= 1

f(xi− 1 )Δxleft-endpoint rectangles

∑n

i= 1

f(xi)Δxright-endpoint rectangles

∑n

i= 1

f

(

xi+xi− 1

2

)

Δxmidpoint rectangles

whereΔx=

b−a

n

anda=x 0 <x 1 <x 2 <···<xn=b.

If f is increasing on [a,b], then left-endpoint rectangles are inscribed rectangles and

the right-endpoint rectangles are circumscribed rectangles. Iffis decreasing on [a,b], then

left-endpoint rectangles are circumscribed rectangles and the right-endpoint rectangles are

inscribed. Furthermore,

∑n

i= 1

inscribed rectangle≤area under the curve≤

∑n

i= 1

circumscribed rectangle.

Example 1

Find the approximate area under the curve of f(x)=x^2 +1 fromx=0tox=2, using

4 left-endpoint rectangles of equal length. (See Figure 13.2-2.)

I II

III

IV

y

(2, 5)

f(x)

x

0 0.5 1 1. 5 2

Figure 13.2-2

LetΔxibe the length ofith rectangle. The lengthΔxi=

2 − 0

4

=

1

2

;xi− 1 =

1

2

(i−1).

Area under the curve≈

∑^4

i= 1

f(xi− 1 )Δxi=

∑^4

i= 1

((

1

2

(i−1)

) 2

+ 1

)(

1

2

)

.

Enter

∑((

(.5(x−1))^2 + 1

)

∗.5,x,1,4

)

and obtain 3.75.