Area Between a Continuous Function and x-Axis

Trip from CC-San Antonio

Make a narrative for the trip

Units of Line and Area Rise/run = miles/hour

Fill region with rectangles

Units of area = length*width = hours*miles

• What is the average velocity (rate of change of distance)?

• What is the average distance from CC during the traveled time?

• What degree polynomial would you use to fit this graph?

GoalFind the area between the graph and the x-axis

Area vs. Absolute AreaArea of a rectangle = length*width or base*height

Base 5, height 1Area = 5Absolute Area =5

Base 5, height -2Area = -10Absolute area= 10

Estimating Areas

• Grid (boxes)• Rectangles• Trapezoids - Boxes-• Indivisibles

An Example to Understand the Techniques

AREA BETWEEN A STRAIGHT LINE AND THE X-AXIS ON A CLOSED INTERVAL

Use basic geometry to calculate this area

Over/Underestimate

Minimum Height

Maximum Height

Minimum height*Width<Area<Maximum Height*Width

Grid (Boxes)

Make a grid. Estimate the number of rectangles needed to fill up the region Estimate area of one rectangleArea No Boxes*Area one box≅

Rectangles (Right/Left)

Height at left hand points Height at right hand points

Left-Hand Sums

Write the summation with 20 subdivisions and use calculator to find the sum.

Right-Hand Sums

Write the summation with 20 subdivisions and use calculator to find the sum.

Trapezoid

area of a trapezoid with heights A, B, and width C is given by (A+B)/2*C

Indivisibles

Indivisible at each end point Average of all IndivisiblesThe area of the rectangle obtained above is the Average

value of the heights multiplied by the length of the interval.

The area between the graph of the continuous function y=f(x)and the x-axis on the interval [a ,b] is denoted

Indivisibles and Average Value of a function

Exercise 2

i) Find an overestimate and underestimate for the ii) Estimate the area using seven subdivisions

a. Grid technique. b. Left-hand rectangles. c. Indivisibles.

For questions (ii) (a-c), also write the expression using summation notation.

Exercise 3

i) Find and overestimate and lower estimate for the area. ii) Estimate the area using rectangles and twelve subdivisions (use summation notation and the calculator).

Exercise 4

Find the area of the region and use it to determine the average distance between the car and CC for

the whole length of the trip.

Area Application #1