The 2nd LiveMath document (IT4AMIntegArea2.thp) shows the
speed-time graph of a car travelling at a speed v = t2 for 4
s. We will use y = x2 for convenience.
2.
To find the area under the curve from x = 0 to x = 4, we use a series
of rectangles of width x
= (b–a)/n and length (or height) y, where a = 0 is the lower
limit, b = 4 is the upper limit
and n is the number of rectangles. Please note that both the limits are
x values, that each rectangle has the same widthx
but different height y, and that bigger n means smaller width x.
Question 3:
The first graph on the left shows the rectangles when n
= 4. Find the width of each rectangle.
Question 4:
Write down the area of each rectangle in terms of x
and y.
Question 5:
The graph shows only 3 rectangles. Where is the 4th one?
Note:
To see the 4th rectangle, change the equation of the curve
to f(x) = x2 + 1. [Place cursor at end of x2
and type + 1.] Change it back before
you proceed.
Question 6:
The 2nd graph in the document shows the rectangles when
n = 8. Find the width of each rectangle.
Question 7:
We use the sum of the areas of the rectangles as an approximation to
the area under the curve. As you can see in the 2 graphs above, the _________
rectangles we have, the better the approximation.
Question 8:
As all these rectangles are under the curve, will the sum of the areas
of the rectangles be greater or less than the actual area under the curve?
The sum of the areas of these rectangles is called
the Lower Riemann Sum.
x
= (b–a)/n and length (or height) y, where a = 0 is the lower
limit, b = 4 is the upper limit
and n is the number of rectangles. Please note that both the limits are
x values, that each rectangle has the same width