Problem 60 - Calculus for Biology and Medicine

Question

In Problems , verify each inequality without evaluating the integrals. $$ \int_{2}^{4} x d x \leq \int_{2}^{4} x^{2} d x $$

Step-by-Step Solution

Answer

Yes, the inequality is verified because \( x \leq x^2 \) for \( x \geq 1 \).

Step 1: Understand the Given Inequality

The inequality to verify is \( \int_{2}^{4} x \, dx \leq \int_{2}^{4} x^{2} \, dx \). This means we need to show that the integral of \( x \) over the interval [2,4] is less than or equal to the integral of \( x^2 \) over the same interval.

Step 2: Compare the Functions

Consider the functions \( f(x) = x \) and \( g(x) = x^2 \). For \( x \in [2, 4] \), it is noticeable that \( f(x) \leq g(x) \) because for any \( x \geq 1 \), \( x \leq x^2 \). This is the comparison which will help us verify the inequality without evaluating the integrals.

Step 3: Use the Property of Integrals

A property of definite integrals states that if \( f(x) \leq g(x) \) for all \( x \) in \([a,b]\), then \( \int_{a}^{b} f(x) \, dx \leq \int_{a}^{b} g(x) \, dx \). Since we established in Step 2 that \( f(x) \leq g(x) \) over \([2, 4]\), it follows that \( \int_{2}^{4} x \, dx \leq \int_{2}^{4} x^{2} \, dx \).

Key Concepts

Definite Integrals

In calculus, a definite integral represents the accumulation of quantities, such as areas under a curve, for a specific interval indicated by the lower and upper summation limits. Understanding definite integrals is crucial since they help calculate the net area between the x-axis and the curve of the function over a specified interval.
\[ \int_{a}^{b} f(x) \, dx \]This notation signifies the definite integral of a function \( f(x) \) from \( a \) to \( b \). Here, \( a \) and \( b \) are the bounds, or limits of integration, and \( f(x) \) is the function being integrated.

• The definite integral measures how much the function accumulates between \( a \) and \( b \).
• Unlike indefinite integrals, definite integrals result in a specific numerical value.
• They can represent areas under curves when \( f(x) \) is positive, but also handle curves below the x-axis by considering areas as negative.

These characteristics make definite integrals essential in solving various real-world problems related to physics, engineering, and economics. It converts a function's rate of change back into accumulated quantity, providing meaningful results under given conditions.

Comparison of Functions

When working with inequalities involving integrals, comparing functions within the same range is a fundamental step. As we saw in our exercise, comparing the functions \( f(x) = x \) and \( g(x) = x^2 \) illustrates this. For any value \( x \geq 1 \), \( x \leq x^2 \), implying \( f(x) \leq g(x)\)in the interval \( [2, 4]\).To compare functions effectively:

• Identify intervals where one function consistently stays above or below the other.
• Determine the function behavior (increasing or decreasing) over the interval.
• Use algebraic inequalities when necessary to confirm comparisons.

In our example, the quadratic formula \( x^2 \) grows more rapidly than the linear function \( x \), ensuring \( x^2 \) is always greater in the interval given. This simple comparison leads to easing the verification of inequalities involving definite integrals.

Properties of Integrals

Definite integrals showcase several properties that simplify calculation and aid in problem-solving. One such essential property is the comparison property, which is critical in verifying integral inequalities. The property asserts:\[ \text{If } f(x) \leq g(x) \text{ for } x \in [a, b], \text{ then } \int_{a}^{b} f(x) \, dx \leq \int_{a}^{b} g(x) \, dx.\]This property comes into play when one can visually or algebraically verify that one function dominates another over a certain interval. Some other notable properties of integrals include:

• Linearity: \( \int_{a}^{b} [cf(x) + dg(x)] \, dx = c\int_{a}^{b} f(x) \, dx + d\int_{a}^{b} g(x) \, dx \). This shows how integrals distribute over addition.
• Reversal of limits: \( \int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx \). Reversing the limits flips the integral's sign.
• Zero Width Interval: \( \int_{a}^{a} f(x) \, dx = 0 \). Integrating over a zero-width interval results in zero.

Understanding these properties allows for solving complex integrations and manipulating inequalities swiftly by estimating or bounding errors within functions.