definite integral

Question

which of the following expressions are true about the definite integral of sinx on the interval [0,2Π]?

 

A) (top 2Π, bottom 0) ∫sinxdx= (Π →∞)lim (top: Π, bottom: i=0)∑sinxi 2Π

 

B) (top 2Π, bottom 0) ∫sinxdx= 0

 

C) (top 2Π, bottom 0) ∫sinxdx= (Π →∞)lim (top: Π, bottom: i=0)∑cosxi-Δx

 

Answer 1

Lets verify answer choice b, since the integral is a short one.

 

∫sin(x)dx = -cos(x)

 

 

Now we include the bounds.

 

-cos(2π) - (-cos(0)) = -1 - (-1)

                          

                             = 1 + 1

                        

                              = 0

 

Answer choice (b) checks out.

Answer 2

The answer is b

 

Since sin function is the complete cycle from 0 to 2pi

0 to pi is positive half cycle.

pi to 2pi is negative half cycle

 

Both are equal and with opposite sign

 

Total area =0

 

So, the integral(sin(x) is =0

 

So, answer is b