1. Collect Homework.
2. Today, we will find areas under curves (i.e., between the curve and the x-axis). We will sum up the areas of rectangles which approximate the area under the curve.
We will use several different techniques:
(1) Using the left-hand endpoints
(2) Using the right-hand endpoints
(3) Using inscribed rectangles
(4) Using the circumscribed rectangles
The figure below shows how to compute the area under the curve by using the left-hand endpoints.
What happens as we increase the number of rectangles?
3. Example:
Determine the area under the curve
y = x + 1 on the interval [0, 2] in three different ways:
(1) Approximate the area by finding areas of rectangles where the height of the rectangle is the y-coordinate of the left-hand endpoint
(2) Approximate the area by finding areas of rectangles where the height of the rectangle is the y-coordinate of the right-hand endpoint
(3) Find the exact area under the curve using geometry
#1 Using left hand endpoints and 4 rectangles: (in this example, these are also called inscribed rectangles)
Area =
(.5)(1) + (.5)(1.5) + (.5)(2) + (.5)(2.5)
= .5 + .75 + 1 + 1.25 = 3.5 square units
#2 Using right hand endpoints and 4 rectangles:
(in this example, these are also circumscribed rectangles)
Area =
(.5)(1.5) + (.5)(2) + (.5)(2.5) + (.5)(3)
= .75 + 1 + 1.25 + 1.5 = 4.5 square units
#3 Using geometry to find the exact area under the curve:
The figure is a trapezoid, so use the formula for the area of a trapezoid:
Area = Average of the Bases x Height or
Note that the average of the first two methods (the inscribed and circumscribed rectangles) gives the exact area under the curve. This only works because the curve is a straight line.
4. Later in the year, we will use other numerical methods to find areas under curves — Simpson's Rule and the Trapezoidal Rule. Simpson's Rule uses parabolas to approximate the area under the curve and the Trapezoidal Rule uses trapezoids to approximate the area.
In general, if we wish to determine the area under the following curve,
we could approximate it with rectangles like we did above or we could approximate it with trapezoids like in the diagram below:
Notice that the trapezoidal approximation gives a more accurate answer than the rectangular approximation.
Tomorrow, we will determine the exact area under the curve by increasing the number of rectangles and then applying a limit.