Given f(x) = -2x + 4.

(a)    Find the Upper and Lower Sum area approximation of f(x) on [0,2] with 5 equal subintervals.
(b)    Calculate the actual area geometrically (note that the shape is a triangle, duh).
(c)    Show that Lower Sum < Actual Area < Upper Sum.
(d)    Find the area on [0,2] using an infinite number of subintervals. Answer should be the same as (b) above.

Solution:

Upper sums are done the same way, but since they are the left hand endpoints of the rectangle intervals (as the above are the right hand endpoints), we replace i with (i-1). The result is 4.8 for that sum. Don't forget that the 2/5 in the above problem comes from the fact that .

(b)    Note that the figure is a triangle with height equal to f(0)=4 and base b-a=2. (1/2)bh is our formula for triangle area, so the actual area is 4.

(c)    Notice that 3.2 < 4 < 4.8.

(d)    We will use the same formula as in part (a) above, except instead of 5, we use n, since we don't know the number of subintervals this time. In fact, we eventually want to let n approach infinity to get the exact area. You'll need to remember a summation formula (the one with just i in it):