$$\int_{\pi/2}^0 \cos x dx = -1$$
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?
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5$\begingroup$ $\int_a^{b}f(x)dx=-\int_b^{a} f(x)dx$ if $a>b$ $\endgroup$– geetha290krmNov 15, 2022 at 11:28
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3$\begingroup$ Does this answer your question? Why does an integral change signs when flipping the boundaries? $\endgroup$– Martin RNov 15, 2022 at 12:31
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1 Answer
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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
$$\int_{\frac\pi2}^0\cos (x)\mathrm dx=-\int_0^{\frac\pi2}\cos (x)\mathrm dx=-1.$$
answered Nov 15, 2022 at 11:35