Problem 70 - Thomas Calculus

Question

Integrals of nonpositive functions Show that if \(f\) is integrable then $$ f(x) \leq 0 \text { on }[a, b] \Rightarrow \int_{a}^{b} f(x) d x \leq 0 $$

Step-by-Step Solution

Answer

If \(f(x) \leq 0\) on \([a, b]\), then the integral \(\int_{a}^{b} f(x) \leq 0\).

Step 1: Understand the Hypothesis

We are given that the function \(f\) is integrable on the interval \([a, b]\) and that for all \(x\) in this interval, \(f(x) \leq 0\). This means that the function takes nonpositive values throughout \([a, b]\).

Step 2: Define the Integral

The integral of \(f\) from \(a\) to \(b\) is defined as \(\int_{a}^{b} f(x)\, dx\). It represents the net area between the graph of \(f\) and the x-axis over the interval \([a, b]\).

Step 3: Apply the Property of Integrals

Since \(f(x) \leq 0\) for all \(x\) in \([a, b]\), it means that the function is either touching or lying below the x-axis. Therefore, each infinitesimal area \(f(x)\, dx\) is nonpositive.

Step 4: Conclude the Integration Result

Summing up all the nonpositive areas from \(a\) to \(b\) results in a nonpositive overall value for the integral. Thus, \(\int_{a}^{b} f(x)\, dx \leq 0\).

Key Concepts

Nonpositive Functions

In mathematics, a function is termed nonpositive if its values are less than or equal to zero for every point in its domain. When dealing with integration, a nonpositive function means that anywhere you look within the interval of interest, the function's values will sit at zero or dip below it.
This concept is crucial because it helps us predict the behavior of integrals. Think of the curve of a nonpositive function as lying at or below the horizontal line representing the x-axis. This scenario implies that all the points in the interval contribute either nothing or negative areas to the total integral.

• If the values are strictly less than zero, the function sits entirely under the x-axis.
• If values are sometimes zero, parts of the function touch the x-axis.
• The combined effect results in a nonpositive or zero net area under the curve.

This nonpositive property simplifies the prediction of integrals as it guarantees outcomes can't be positive.

Definite Integration

Definite integration is a process in calculus used to calculate the accumulation of quantities, which can be thought of as the net area under the curve of a function. The "definite" aspect comes from the specific boundaries over which this calculation occurs, namely \[ \int_{a}^{b} f(x) \, dx, \] where \(a\) and \(b\) are the limits of integration.
In the context of nonpositive functions, the process remains the same, but the interpretation differs slightly:

• If the function is nonpositive, the integral calculates the net negative area due to the function lying beneath the x-axis.
• This results in the integral value being nonpositive.
• The boundaries are important as they establish the start and end points for the net area calculation.

The beauty of definite integration lies in its applicability across various scientific fields, enabling the calculation of quantities like areas, distances, and more.

Properties of Integrals

Understanding the properties of integrals is vital to mastering calculus concepts. These properties can simplify calculations and provide insights into the behavior of integrals. A key property to consider, especially when dealing with nonpositive functions, is that of sign preservation:

• If \(f(x) \leq 0\) over an interval \([a, b]\), then \(\int_{a}^{b} f(x) \, dx \leq 0\).
• This stems from the fact that negative inputs across an interval result in a nonpositive sum.

Another important property is linearity, which allows integrals to be added or subtracted, and constants to be factored out:

\[\int_{a}^{b} (cf(x) + g(x)) \, dx = c \int_{a}^{b} f(x) \, dx + \int_{a}^{b} g(x) \, dx\]

• This property is useful when breaking down complex functions into simpler parts.
• For nonpositive functions, linearity can simply confirm that a linear combination of such functions still yields a nonpositive integral if the coefficients are positive.

These are just a few examples, but properties of integrals act as powerful tools in simplifying and solving integrals.