Why is $\int_{\pi/2}^0 \cos x dx$ negative?

Question

$$\int_{\pi/2}^0 \cos x dx = -1$$

But we know that integration is an area under the curve.

$\cos x$ is positive between $0$ and $\pi/2$. So, the area should be positive. But why is the integration negative?

Answer 1

The integral is a directed area. And by swapping the lower and upper limit, you change the direction of every area. Which is why you have to invert the sign of the integral if you do. So we have

$$\int_{\frac\pi2}^0\cos (x)\mathrm dx=-\int_0^{\frac\pi2}\cos (x)\mathrm dx=-1.$$