5 Steps to a 5 AP Calculus BC 2019

Areas, Volumes, and Arc Lengths 263

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 12.2-2.)

I II

III

IV

y

(2,5)

f(x)

0 0.5 1 21.5 x

Figure 12.2-2

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
∑((
( 0 .5(x−1))^2 + 1

)

∗ 0 .5,x,1,4

)

and obtain 3.75.