Area Between a Continuous Function and x-Axis

Trip from CC-San Antonio Make a narrative for the trip/Average velocity

Units of Line and Units of Area Rise/run = miles/hour Fill region with rectangles to estimate area Units of area = length*width = hours*miles Units of area = length*width = hours*miles

What is the average velocity (rate of change of distance) for the whole trip? What is the average distance from the car to CC during the traveled time? What degree polynomial would you use to fit this graph? Questions to discuss

Goal Find the area between the graph and the x-axis

Nice Examples where Area is needed Bicycle 1 Do rides 1, 3, 5 Bicycle 1 Bicycle 2

Area vs. Absolute Area Area of a rectangle = length*width or base*height Base 5, height 1 Area = 5 Absolute Area =5 Base 5, height -2 Area = -10 Absolute 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 rectangle Area ≅ 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 Indivisibles The 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 Intuition about the geometry of integrals

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