The value of a definite integral will always be positive. ? True ? False
The Correct Answer and Explanation is:
Correct Answer: ❌ False
Explanation:
The value of a definite integral is not always positive. A definite integral represents the net area between the graph of a function and the x-axis over a given interval [a,b][a, b][a,b]. This net area can be positive, negative, or zero, depending on the behavior of the function within that interval.
Positive vs. Negative Area:
- If the function f(x)f(x)f(x) is entirely above the x-axis on the interval [a,b][a, b][a,b] (i.e., f(x)≥0f(x) \geq 0f(x)≥0 for all xxx in that interval), then the definite integral ∫abf(x) dx\int_a^b f(x) \, dx∫abf(x)dx will be positive or zero.
- If the function f(x)f(x)f(x) is entirely below the x-axis on the interval (i.e., f(x)≤0f(x) \leq 0f(x)≤0), then the integral will be negative or zero.
- If the function crosses the x-axis (i.e., takes on both positive and negative values), the integral adds up positive and negative contributions, resulting in the net area, which could be positive, negative, or zero.
Example:
Consider the function f(x)=−2f(x) = -2f(x)=−2 over the interval [0,3][0, 3][0,3]:∫03−2 dx=−2⋅(3−0)=−6\int_0^3 -2 \, dx = -2 \cdot (3 – 0) = -6∫03−2dx=−2⋅(3−0)=−6
Here, the function is constant and negative, so the integral yields a negative value.
Clarification:
If you are only interested in the total area regardless of whether it’s above or below the x-axis, you would use the absolute value:∫ab∣f(x)∣ dx\int_a^b |f(x)| \, dx∫ab∣f(x)∣dx
This always gives a positive result, but it’s a different kind of integral with a different meaning.
Conclusion:
The value of a definite integral depends on the sign of the function within the interval. Hence, the statement “The value of a definite integral will always be positive” is false.
