Calculus integrals area under curve and average value

Question

 

 

I need help with both of the following questions. Can someone give me an explination of how to calculate each of these:

 

A. Calculate the area of the region under the curve 4e^(2x) between x = -0.5 to x =0.5 .

B. Find the average value of the function in the same interval.

Answer 1

simply integrate 4e^2x dx over the period you get the area as

2e^2x (x = -0.5 to 0.5) = 2(e-1/e) = 2(e^2-1)/e

the average is the same as the area because the interval is of length 1