Calculus
Lesson 21 Area
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The entire area is the sum of all of the smaller areas!
where each little area is 
for lower sum
and 
for
upper sum.
Let's find the area under 
on
the interval 
, using 4
rectangles.
Draw Separate Picture for Each!!
Upper Sum:
Lower Sum:
Now for 8 rectangles:
Upper Sum: 
Lower sum:
Then with n rectangles Upper Sum: 
Then with n rectangles Lower Sum:
What happens as 
? Upper Sum:
Lower Sum:
Note that the sums are the same!!!!!!!!
Our definition for area under a curve, then,
should be: 
Practice writing sums, using various functions. Also practice evaluating sums by hand and by calculator: sum( is 2nd,list,math: 0.25 * sum(seq(x^2,x,0,2,.25)) is 8 rectangle upper sum. Or make a list of x^2 values in L2 and answer is 0.25*sum(L2).
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