Find the area of the region bounded by the graphs of $y = 1$ and $y = \cos^2(x)$ from $x = 0$ to $x = π$

Question

Find the area of the region bounded by the graphs of $y = 1$ and $y = \cos^2(x)$ from $x = 0$ to $x = π$.

$a. 0.785$

$b. 3.142$

$c. 2.576$

$d. 1.571$

I graphed it but I'm not sure what to do next. Please help

Answer 1

When you mean $\cos^2\left(x\right)$:

$$\int_0^\pi1\space\text{d}x-\int_0^\pi\cos^2\left(x\right)\space\text{d}x=\frac{\pi}{2}\approx1.570796326794897$$

Answer 2

Use the double-angle formula:

$$ cos (2x) = cos^2(x) - sin^2(x)$$ $$ 1 = cos^2(x) + sin^2(x)$$

Add those two equations and you get $cos(2x) + 1 = 2 cos^2(x)$

Divide by 2 and you get the handy reduction formula

$$cos^2(x)= \frac{cos(2x) + 1}{2} $$

Substitute this in your integral and with a simple substitution you can do the rest of the integral.