AP Calculus

Area

Area of a Plane Region

• Calculus was built around two problems

– Tangent line

– Area

Area

• To approximate area, we use rectangles

• More rectangles means more accuracy

Area

• Can over approximate with an upper sum

• Or under approximate with a lower sum

Area

• Variables include

– Number of rectangles used

– Endpoints used

Area

• Regardless of the number of rectangles or types of inputs used, the method is basically the same.

• Multiply width times height and add.

Upper and Lower Sums

• An upper sum is defined as the area of circumscribed rectangles

• A lower sum is defined as the area of inscribed rectangles

• The actual area under a curve is always between these two sums or equal to one or both of them.

Area Approximation

• We wish to approximate the area under a curve f from a to b.

x

• We begin by subdividing the interval [a, b] into n subintervals.

• Each subinterval is of width .

Area Approximation

ba

f

Area Approximation

• We wish to approximate the area under a curve f from a to b.

n

abx

• We begin by subdividing the interval [a, b] into n

subintervals of width .

i

i

Mf

mf Minimum value of f in the ith subintervalMaximum value of f in the ith subinterval

Area Approximation

ba

f

x

i

i

mf

Mf

Area Approximation

• So the width of each rectangle is n

abx

ba

f

x

i

i

mf

Mf

• So the width of each rectangle is n

abx

• The height of each rectangle is either imf

or iMf

Area Approximation

ba

f

x

i

i

mf

Mf

Area Approximation

• So the width of each rectangle is

xMfnSn

ii

1

n

abx

• The height of each rectangle is either imf

or iMf• So the upper and lower sums can be defined as

xmfnsn

ii

1

Lower sum

Upper sum

Area Approximation

• It is important to note that

• Neither approximation will give you the actual area

• Either approximation can be found to such a degree that it is accurate enough by taking the limit as n goes to infinity

• In other words

nSAreans )(

nSnsnn

limlim