Question

Prove: If \(f\) is integrable and \(f(x) \geq 0\) on \([a, b]\), then \(\int_{a}^{b} f(x) d x \geq 0,\) with strict inequality if \(f\) is continuous and positive at some point in \([a, b]\).

Short answer

Question: Prove that if a function \(f(x)\) is integrable on the interval \([a, b]\) and \(f(x) \geq 0\) for all \(x\) in \([a, b]\), then the integral of \(f(x)\) is also greater than or equal to 0. Additionally, if \(f(x)\) is continuous and positive at some point in \([a, b]\), show that the integral has a strict inequality, i.e., it's greater than 0.

Step-by-step solution

Prove that integration preserves non-negativity

Since \(f(x) \geq 0\) for all \(x\) in \([a, b]\), the area under the curve will be positive or equal to zero. The Riemann integral represents the area under the curve, so when we sum up all the small rectangles, we can say that \(\int_{a}^{b} f(x) dx = A \geq 0\).

Prove strict inequality if f is continuous and positive at some point

Assume that \(f(x)\) is continuous and positive at a point \(c\) in \([a, b]\), meaning that \(f(c) > 0\). Since \(f(x)\) is continuous, there exists a small interval around the point \(c\), say \([c-\delta, c+\delta]\) with \(\delta > 0\), such that \(f(x) > \frac{1}{2}f(c)\) for all \(x\) in the interval, by the continuity of \(f(x)\). Now consider the integral of \(f(x)\) over this small interval: \(\int_{c-\delta}^{c+\delta} f(x) dx \geq \int_{c-\delta}^{c+\delta} \frac{1}{2}f(c) dx\) Evaluating the integral on the right-hand side: \(\frac{1}{2}f(c)\int_{c-\delta}^{c+\delta} dx = \frac{1}{2}f(c)(2\delta) = f(c)\delta > 0\) Since we have shown that there is a positive contribution from the interval \([c-\delta, c+\delta]\), this implies that the overall integral is greater than 0: \(\int_{a}^{b} f(x) dx > \int_{c-\delta}^{c+\delta} f(x) dx > 0\) Thus, we have shown that if \(f(x)\) is continuous and positive at some point in \([a, b]\), then the integral is strictly greater than 0.

Key concepts

Integration

Integration is the process of finding the whole from its parts. In mathematics, it means summing up values to find the area under a curve. This area represents the integral of a function. When we integrate a function like \( f(x) \) over an interval \([a, b]\), we are essentially calculating the total accumulation of \( f \) between \( a \) and \( b \).

• The Riemann Integral is a method to approximate this area by dividing it into thin rectangles.
• The width of each rectangle is very small, and their sum gives us the integral value.
• If \( f(x) \) is non-negative, the rectangles will have non-negative heights, so the area is not less than zero.

Understanding the integral as the area makes it easy to see why if \( f(x) \geq 0 \), then \( \int_{a}^{b} f(x) \, dx \geq 0 \). There is no negative area, ensuring non-negativity.

Continuity

Continuity of a function means that it doesn't have any breaks, jumps, or holes over its domain. When a function \( f(x) \) is continuous, it smoothly transitions from one point to another.

• Continuous functions are predictable and steady over any interval \([a, b]\).
• This property is crucial when proving strict inequalities for integrals.
• If \( f(x) \) is continuous and positive at a point \( c \), a small interval around \( c \) will also have positive values.

By ensuring that \( f(x) \) is never zero around such points, we secure a contribution that makes the integral strictly greater than zero, highlighting the power of continuity in integration.

Non-negative Functions

Non-negative functions have values greater than or equal to zero over their entire domain. This simple property has profound implications for integrals.

• If \( f(x) \geq 0 \) for every \( x \) in \([a, b]\), then all contributions to the integral are non-negative.
• The Riemann sum, which approximates the integral, will also be composed of non-negative terms.
• This ensures \( \int_{a}^{b} f(x) \, dx \geq 0 \), as subtracted negative parts would only reduce the area.

The interesting part comes when \( f \) is not just non-negative but also strictly positive at some point; combined with continuity, this positivity ensures the integral is greater than zero, moving from equality to strict inequality.