If \( \int_{3}^{7} f(x) d x=12 \), ther \( \int_{7}^{3} f(x) d x=[?] \)

Did you know that reversing the limits of integration affects the sign of the integral? So, if you have \( \int_{3}^{7} f(x) \, dx = 12 \), then simply flipping the limits gives you \( \int_{7}^{3} f(x) \, dx = -12 \). It's like taking a scenic journey but in the opposite direction! This principle comes in handy when calculating definite integrals quickly without needing to recompute the entire area under the curve. Whenever you see a negative sign in your limits, just remember it’s the opposite of the original integral value!