Question
If \( \int_{3}^{7} f(x) d x=12 \), ther \( \int_{7}^{3} f(x) d x=[?] \)
Ask by Christensen Ingram. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\[
\int_{7}^{3} f(x) \, dx = -12
\]
Solution
The integral with reversed limits is the negative of the original integral. In other words:
∫₇³ f(x) dx = -∫₃⁷ f(x) dx
Since ∫₃⁷ f(x) dx = 12, it follows that:
∫₇³ f(x) dx = -12.
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Simplify this solution Extra Insights
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!
