Problem 60 - Calculus

Question

If \(f\) and \(g\) are continuous and \(f(x)>g(x)\) for all \(x\) in \([a, b],\) then \(\left|\int_{a}^{b} f(x) d x\right|>\left|\int_{a}^{b} g(x) d x\right|\)

Step-by-Step Solution

Answer

\( \left|\int_a^b f(x) \, dx\right| > \left|\int_a^b g(x) \, dx\right| \) because \( f(x) > g(x) \) for all \( x \) in \([a, b]\).

Step 1: Understanding the Problem

We are given two continuous functions, \( f(x) \) and \( g(x) \), with the condition \( f(x) > g(x) \) for all \( x \) in the interval \([a, b]\). We need to prove that the absolute value of the integral of \( f(x) \) over \([a, b]\) is greater than the absolute value of the integral of \( g(x) \) over the same interval.

Step 2: Analyze the Integrals

Consider the integrals \( \int_a^b f(x) \, dx \) and \( \int_a^b g(x) \, dx \). Since \( f(x) > g(x) \) for all \( x \) in \([a, b]\), this suggests that the area under curve \( f(x) \) is greater than the area under curve \( g(x) \).

Step 3: Using the Property of Integrals

The property \( f(x) > g(x) \) for all \( x \) in \([a, b]\) allows us to state that \( \int_a^b f(x) \, dx - \int_a^b g(x) \, dx > 0 \). This is because the integrand \( f(x) - g(x) \) is positive over the entire interval \([a, b]\).

Step 4: Conclusion from the Integral Property

Since \( \int_a^b f(x) \, dx - \int_a^b g(x) \, dx > 0 \), then \( \int_a^b f(x) \, dx > \int_a^b g(x) \, dx \). Hence, the absolute value \( \left|\int_a^b f(x) \, dx\right| = \int_a^b f(x) \, dx \) is greater than \( \left|\int_a^b g(x) \, dx\right| = \int_a^b g(x) \, dx \).

Key Concepts

Continuous Functions

Continuous functions are essential in calculus since they offer a stable behavior over an interval. When a function is continuous over an interval

• There are no breaks, jumps, or holes in its graph within that interval.
• The function is smooth and predictable, making it easier to work with analytically.

For the exercise, having continuous functions, like \(f(x)\) and \(g(x)\), is crucial as it guarantees the existence of their integrals over \([a, b]\). This is because the Fundamental Theorem of Calculus applies seamlessly to continuous functions, ensuring we can compute the integral accurately. By being continuous on \([a, b]\), \(f(x)\) and \(g(x)\) are integrable on this interval, setting the stage for evaluating their integrals with confidence.
Thus, continuity helps to ensure that the process of integration reflects the true area under the curves of these functions.

Definite Integrals

Definite integrals play an integral role in finding the accumulated values over a certain interval. They are often used to calculate areas under curves, the net change, or total accumulation.
In mathematical terms, the definite integral of a function \(f\) over an interval \([a, b]\) is expressed as follows:\[\int_a^b f(x) \, dx \]
This specific integral provides the signed area under the curve of \(f(x)\) from \(a\) to \(b\).
It's worth noting:

• The result of the definite integral is a real number, not a function.
• If \(f(x)\) is above the x-axis, the integral gives the area; if below, it subtracts the area.

In the exercise, assessing the integrals \(\int_a^b f(x) \, dx\) and \(\int_a^b g(x) \, dx\) involves comparing these accumulated areas between two functions where \(f(x) > g(x)\). This ensures that the integral of \(f\) over the interval is larger than that of \(g\), confirming the area under \(f(x)\) is greater.

Properties of Integrals

The properties of integrals are pivotal in solving inequalities involving integrals, like in this exercise. One of the primary properties used is the linearity of integration.
This property states that for any two functions \(f(x)\) and \(g(x)\):\[\int_a^b (f(x) - g(x)) \, dx = \int_a^b f(x) \, dx - \int_a^b g(x) \, dx\]
Through this, we recognize:

• If \(f(x)\) is always greater than \(g(x)\) over \([a,b]\), then the difference \((f(x) - g(x))\) is positive.
• Hence, the integral of this difference over \([a,b]\) is greater than zero.

This is critical as it builds our understanding that \(\int_a^b f(x) \, dx > \int_a^b g(x) \, dx\). In mathematical proofs or solving inequalities like this, properties such as this allow us to leverage comparisons.
These properties help us deduce or infer other results based on known information, enhancing both understanding and calculation accuracy.