Problem 37 - Essential Calculus

Question

Given \(\int_{2}^{6} f(x) d x=10\) and \(\int_{2}^{6} g(x) d x=-2,\) evaluate (a) \(\int_{2}^{6}[f(x)+g(x)] d x\). (b) \(\int_{2}^{6}[g(x)-f(x)] d x\). (c) \(\int_{2}^{6} 2 g(x) d x\). (d) \(\int_{2}^{6} 3 f(x) d x\).

Step-by-Step Solution

Answer

(a) The value of the integral is 8.\n(b) The value of the integral is -12.\n(c) The value of the integral is -4.\n(d) The value of the integral is 30.

Step 1: Applying linearity of integral for (a)

Property of linearity allows the integrand to be split into the sum of two integrals: \(\int_{2}^{6}[f(x)+g(x)] d x = \int_{2}^{6} f(x) d x + \int_{2}^{6} g(x) d x = 10 - 2 = 8.\)

Step 2: Applying linearity of integral for (b)

Similarly, the integrand can be split into the difference of two integrals: \(\int_{2}^{6}[g(x)-f(x)] d x = \int_{2}^{6} g(x) d x - \int_{2}^{6} f(x) d x = -2 - 10 = -12.\)

Step 3: Applying linearity of integral for (c)

The linearity property also applies to the scale in front of the function in the integral: \(\int_{2}^{6} 2 g(x) d x = 2 \cdot \int_{2}^{6} g(x) d x = 2 \cdot (-2) = -4.\)

Step 4: Applying linearity of integral for (d)

And, finally, for a scaling inside the f(x) function the property looks like this: \(\int_{2}^{6} 3 f(x) d x = 3 \cdot \int_{2}^{6} f(x) d x = 3 \cdot 10 = 30.\)

Key Concepts

Integral Calculation

Integral calculation is essential in determining the area under a curve for a given function over a specific interval. It works by summing up infinitesimally small areas under the curve, to give us a total value. This concept plays a crucial role in both mathematics and applied sciences because it allows us to find quantities like accumulated change or total area.
In the exercise provided, integral calculation focuses on finding the result of various integrals formed by combining functions. Each part of the exercise uses the principle of adding integrals of simpler functions to calculate more complex ones. This method simplifies a potentially complex calculation by breaking it into manageable parts.

• Use given individual integrals to construct new integrals.
• Exploit the properties of linearity to simplify the calculations.
• Strategically combine and manipulate integrals to achieve a desired result.

Integral calculation becomes much easier when you apply these properties, and it helps students confidently solve complex integral problems.

Definite Integrals

Definite integrals represent the accumulation of a function across a specified range, from lower limit to upper limit. It is denoted as \(\int_{a}^{b} f(x) \, dx\) where \(a\) and \(b\) are the limits of integration. This form of integral provides a specific numerical value, as opposed to indefinite integrals, which express an accumulation function without specific bounds.
In our exercise, we looked at definite integrals over the interval \([2, 6]\). Calculating these integrals tells us the net area between the function curve and the x-axis from \(x = 2\) to \(x = 6\). In practice, this means the exercise already has the base integrals calculated for \(f(x)\) and \(g(x)\), making it easier to explore different combinations.

• Example: \(\int_{2}^{6} f(x) \, dx = 10\).
• Definite integrals help in understanding the net change over a specific interval.
• Crucial for problems that involve precise measurements over a particular range.

This demonstrates how useful definite integrals are in giving real-world meaning to changes over intervals.

Linear Combination of Functions

A linear combination of functions involves adding or subtracting functions, each possibly multiplied by a constant, to form a new function. Mathematically, it appears as a mix, like \(af(x) + bg(x)\), where \(a\) and \(b\) are constants.
In the original exercise, various linear combinations of \(f(x)\) and \(g(x)\) were presented to explore the property of linearity in integrals. This principle states that the integral of a linear combination of functions is equal to the linear combination of their integrals.

• Example: \(\int_{2}^{6} [f(x) + g(x)] dx = \int_{2}^{6} f(x) dx + \int_{2}^{6} g(x) dx\).
• Linearity simplifies calculations by allowing separate integration of each component function.
• Scaling of functions (e.g., \(3f(x)\) or \(2g(x)\)) affects the integral result proportionally.

Understanding linear combinations allows students to effortlessly tackle integration problems by transforming them into simpler parts.