Before integration was developed, people found the area under curves by dividing the space beneath into rectangles, adding the area, and approximating the answer.

Right Riemann places the right point of the rectangles along the curve to find the area. The equation that is used for the RIGHT RIEMANN ALWAYS begins with:

Given this problem below, what all do we need to know in order to find the area under the curve using Right Riemann?

4 partitions

Right Riemann- Example

For each method we must know:

f(x)- the function of the curve

n- the number of partitions or rectangles

(a, b)- the boundaries on the x-axis between which we are finding the area

Right Riemann- Example

Right Riemann TRY ME!

Volunteer:___________________

4 Partitions

!Show All Your Work!

n=4

Did You Get It Right?

n=4

Left Riemann- Overview

Left Riemann uses the left corners of rectangles and places them along the curve to find the area. The equation that is used for the LEFT RIEMANN ALWAYS begins with:

And ends with

Within the brackets!

Left Riemann- Example

Remember: Left Only

Given this problem below, what all do we need to know in order to find the area under the curve using Left Riemann?

4 partitions

Left Riemann- Example

For each method we must know:

f(x)- the function of the curve

n- the number of partitions or rectangles

(a, b)- the boundaries on the x-axis between which we are finding the area

Left Riemann- Example

Left Riemann- TRY ME!

Volunteer:___________

3 Partitions

!Show All Your Work!

n=3

Did You Get My Answer?

n=3

Midpoint Riemann- Overview

Midpoint Riemann uses the midpoint of the rectangles and places them along the curve to find the area. The equation that is used for MIDPOINT RIEMANN ALWAYS begins with:

And ends with

Within the brackets!

Midpoint Riemann- Example

Remember: Midpoint Only

Given this problem below, what all do we need to know in order to find the area under the curve using Midpoint Riemann?

4 partitions

Midpoint Riemann- Example

For each method we must know:

f(x)- the function of the curve

n- the number of partitions or rectangles

(a, b)- the boundaries on the x-axis between which we are finding the area

Trapezoidal Rule is a little more accurate that Riemann Sums because it uses trapezoids instead of rectangles. You have to know the same 3 things as Riemann but the equation that is used for TRAPEZOIDAL RULE ALWAYS begins with:

Given this problem below, what all do we need to know in order to find the area under the curve using Trapezoidal Rule?

4 partitions

Trapezoidal Example

For each method we must know:

f(x)- the function of the curve

n- the number of partitions or rectangles

(a, b)- the boundaries on the x-axis between which we are finding the area

Trapezoidal Rule- Example

Trapezoidal Rule- TRY Me

Volunteer:_____________

4 Partitions

Trapezoidal Rule- TRY ME!!

n=4

Was this your answer?

n=4

Simpson’s Rule- Overview

Simpson’s rule is the most accurate method of finding the area under a curve. It is better than the trapezoidal rule because instead of using straight lines to model the curve, it uses parabolic arches to approximate each part of the curve. The equation that is used for Simpson’s Rule ALWAYS begins with:

And ends with

Within the brackets with every “f” being multiplied by alternating coefficients of 4 and 2 EXCEPT the first and last terms.

In Simpson’s Rule, n MUST be even.

Simpson’s Rule- Example

Remember: Simpson’s Rule Only

Given this problem below, what all do we need to know in order to find the area under the curve using Simpson’s Rule?

4 Partitions

Simpson’s Example

For each method we must know:

f(x)- the function of the curve

n- the number of partitions or rectangles

(a, b)- the boundaries on the x-axis between which we are finding the area