5.1 Sequences and Series by SGLee - HSKim- SWSun
1. Find the area under the curve from 0 to 2.
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.
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.