Q.If $\displaystyle\int_{2}^{7} f(x)\,dx = 15$, what is the value of $\displaystyle\int_{7}^{2} f(x)\,dx$?

Reversing the limits of a definite integral changes only its sign, never its magnitude.

This question tests the property that reversing the limits of a definite integral changes the sign of its value:

Here $a=2$, $b=7$, and $\int_{2}^{7}f(x)\,dx=15$ is given, so:

Why the other options are wrong:

• (a) $15$ — this is the value of $\int_{2}^{7}f(x)\,dx$ itself, obtained by ignoring the limit reversal entirely.
• (c) $0$ — this would be correct only if the two limits were EQUAL (the zero-width property, $\int_{a}^{a}f(x)\,dx=0$), which is not the case here since $7\ne2$. …

Assuming reversing the limits leaves the integral unchanged (choosing option (a)), or confusing the sign-reversal property with the separate zero-w …