Which of these integrals is larger, why?

Question

The integrals are 

a) ∫(.5,0) cos(x^2) dx (consider .5 the top number for the integral and 0 the bottom number.)  

b) ∫(.5, 0) cos(x^1/2 ) dx

Answer 1

The integral is the area under the curve between the curve and x-axis.  In both cases, the bounds are the same.  So we just need to compare the functions that associated with the area.

 

Notice that the functions are cosine, but the angles are different.

 

For a)

 

cos((1/2)²) = cos(1/4) = .99999048

 

 

For b)

 

cos(√(1/2)) = cos(1/√2) = .9999238

 

 

The square has the greater output.  This will allow for greater area under curve.  Therefore, (a) is the larger integral.