Let f (x) = cos x − sin x for 0 ≤ x ≤ 𝜋/2. Find the area A of the region between the graph of f and the x-axis on the given interval

Question

(Hint: Where is f (x) > 0 and where is f (x) < 0?)

Answer 1

To find area between the x-axis and a function, the integral is used:

pi/2

∫[cos(x) - sin(x)]dx

0

integral of cos(x) and sin(x) are reoccurring and follow this rule I've memorized:

-c -> s -> c -> -s

Where taking the derivative goes toward the right, and integrating goes toward the left (of only the sine and cosine functions representing s and c respectively)

So that would mean the above function is:

∫f(x)dx = g(x) = - sin(x) - cos(x) from the limits of 0 to pi/2

g(0) = - [sin(0) + cos(0)] = -1

g(pi/2) = - [sin(pi/2) + cos(pi/2)] = -1

sum them: -1 + -1 = -2