The Area Under a Curve5.1

Sigma Notation ∑ is the Greek letter sigma

means to plug every integer from

1 through n in for i and add the results

For example:

1

n

i

i

6

1

1 2 3 4 5 6 21i

i

The Area Under a Curve The next topic in calculus is finding

the area under a curve. Area under a curve – the area

between the graph of a curve and the x axis.

Positive area is above the x axis and negative area is below the x axis.

The Area Under a Curve This is a graph of The area under the curve from a to b is shaded.

How can we approximate this area? a

b

2y x

Rectangular Approximation Method (RAM)

Break the area into equal Rectangles to approximate the area. N designates the number of rectangles we will use.

Δx is the width of the rectangles. a is the lower limit of the area (x

value) b is the upper limit of the area (x

value)

b axn

Three Methods of RAM Right hand RAM (RRAM)

Three methods of RAM Left hand RAM (LRAM)

Three Methods of RAM Middle RAM (MRAM)

Example RRAM Approximate the area under the

curve of from 1 to 3 using RRAM and n =

6

2y x

Example LRAM Approximate the area under the

curve of from 1 to 3 using LRAM and n =

6

2y x

Example MRAM Approximate the area under the

curve of from 1 to 3 using MRAM and n =

6

2y x

Homework Use all three methods and n=6 to

approximate the area under the curve

on the interval [0,2]

Use any one method and n = 4 to find the area under the curve on the interval [0, π]

3y x

siny x