Calculus-Sec-5-1-Solution
5.1 Sequences and Series by SGLee - HSKim- SWSun
1. Find the area under the curve 
from 0 to 2.
http://matrix.skku.ac.kr/cal-lab/cal-5-1-exs-1.html
We divide the interval [0,2] into n equal parts.
Thus the length of each sub-interval is 2/n and the 
th sub-interval is given by 
Now we apply the right end formula to find required area.
.
2. Find the area of the region under the graph of 
from 0 to 2.
http://matrix.skku.ac.kr/cal-lab/cal-5-1-exs-2.html
.
3. Find the area under the curve 
from 
to 
, where 
.
. (Since 
, the value what we evaluate is equal to the area.)
4. (a) Let 
be the area of a polygon with 
equal sides inscribed in a circle with radius 
.
By dividing the polygon into 
congruent triangles with central angle 
,
show that 
.
(b) Show that 
.
[Hint: Use Equation 3.4.2]
(a) The area of one piece of 
consists of 
such pieces so the area of 
Remark: To estimate area under the graph of 
, one can take the sample point 
as the point 
such that 
, 
.
Similarly we can choose the sample point 
as the point 
such that 
.
(b) Using 
,
(Let 
. Then 
implies 
.) Think of the lower and upper sums of the graph.
