Calculus Question

Question

What is the area of the region that is below both of the curves y = sinx and y = cosx but above the x axis from x=0 to x=π/2?

a.1/2

b.√(2)

c.π/4

d. 2-√(2)

e.1

f.π/3

Answer 1

Draw both curves between x = 0 and x = π.

The desired region consists of two parts: The region below the graph of y = sinx and above the x-axis between x = 0 and x = π/4, and the region below the graph of y = cosx and above the x-axis between x = π/4 and x = π/2.

Area = ∫_{from 0 to} π_/4 (sinx)dx + ∫_{from π/4 to π /2} (cosx)dx

= -(cosπ/4 - cos0) + (sinπ/2 - sinπ/4)= -√2/2 + 1 + 1 - √2/2 = 2 - √2