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.

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.