Problem 41 - Calculus Early Transcendentals

Question

Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{-2}^{-1} x^{-3} d x$$

Step-by-Step Solution

Answer

Question: Evaluate the integral $$\int_{-2}^{-1} x^{-3} d x$$. Answer: $$\int_{-2}^{-1} x^{-3} d x = \frac{-3}{8}$$

Step 1: Find the antiderivative, F(x)

To find the antiderivative of $$f(x) = x^{-3}$$, use the power rule for integration, which states that $$\int x^n d x = \frac{x^{n+1}}{n+1}+ C$$, where n is a constant and C is the integration constant: $$\int x^{-3} d x = \frac{x^{-3+1}}{-2} + C = \frac{x^{-2}}{-2} + C$$ Therefore, the antiderivative of $$f(x) = x^{-3}$$ is F(x) = $$-\frac{1}{2}x^{-2} + C$$.

Step 2: Apply the Fundamental Theorem of Calculus

Now, we have the antiderivative F(x), and we can use it to evaluate the integral $$\int_{-2}^{-1} x^{-3} d x$$ according to the Fundamental Theorem of Calculus: $$\int_{-2}^{-1} x^{-3} d x = \left[-\frac{1}{2}x^{-2}\right]_{-2}^{-1}$$ $$= -\frac{1}{2}(-1)^{-2} - \left(-\frac{1}{2}(-2)^{-2}\right)$$

Step 3: Simplify and find the final result

Now, simplify the expression and find the final result: $$= -\frac{1}{2}(1) - \left(-\frac{1}{2}\times \frac{1}{4}\right)$$ $$= -\frac{1}{2} + \frac{1}{8}$$ $$= \frac{-4 + 1}{8}$$ $$= \frac{-3}{8}$$ So, $$\int_{-2}^{-1} x^{-3} d x = \frac{-3}{8}$$.

Key Concepts

Integration

Integration is a fundamental concept in calculus that represents the process of finding the area under a curve. In simpler terms, if we're trying to find out how much space lies beneath a graph and above the x-axis, integration is the tool we use. It's like adding up infinitely many thin slices to find a total quantity. For example, suppose we are interested in the cumulative effect of a changing quantity, such as the total distance traveled by a car with changing speed, integration can tell us that total distance.

When we write an integral, such as \[\int_{a}^{b} f(x) dx\], we're outlining a few things. The function \(f(x)\) is what we're integrating, and the letters \(a\) and \(b\) are the bounds of our integral – the start and the end of the region we're examining. The \(dx\) indicates that we're integrating with respect to \(x\) – that is, we're looking at how our function changes as \(x\) changes over the interval from \(a\) to \(b\).

Integration is essential in various fields, including physics, engineering, economics, and statistics, as it allows us to work out problems related to rates of change and accumulation.

Antiderivative

An antiderivative, also known as an indefinite integral, can be thought of as the opposite of differentiation. If differentiation gives us the rate at which a function is changing, then an antiderivative provides us with the original function before it was differentiated. Mathematically speaking, if \(F(x)\) is an antiderivative of \(f(x)\), then \(F'(x) = f(x)\). This means that when you take the derivative of \(F(x)\), you get back to the original function \(f(x)\).

In the context of the fundamental theorem of calculus, the antiderivative plays a crucial role. This theorem bridges the gap between antiderivatives and definite integrals, allowing us to evaluate the area under a curve by simply knowing the values of an antiderivative at the bounds of integration. It essentially tells us that integration and differentiation are inverse processes.

For example, in the provided exercise, the antiderivative of \(x^{-3}\) is found using the power rule for integration, which we can then use to evaluate the definite integral over a particular interval. It's important to remember that when finding an antiderivative, we always add a constant of integration (\(C\)) because derivatives of constants are zero and thus, many functions can have the same derivative.

Power Rule for Integration

The power rule for integration is a quick way to find the antiderivative of a function in the form of \(x^n\), where \(n\) is a real number and not equal to -1. The rule states \[\int x^n dx = \frac{x^{n+1}}{n+1} + C\], where \(C\) is the integration constant. The power rule is derived from the concept that when we differentiate a power of \(x\), we lower the exponent by one and multiply by the old exponent.

The steps to apply the power rule for integration are as follows:

• Identify the exponent \(n\) in your function.
• Add 1 to the exponent to get \(n+1\).
• Divide the term by \(n+1\).
• Add the constant of integration, \(C\).

For negative or fraction exponents, the same process is used, just with extra care for handling the negative or fractional values. However, be cautious when the exponent \(n = -1\); in this case, the power rule can't be applied, and the antiderivative is the natural logarithm function instead. In the example given in the exercise, the function \(x^{-3}\) is integrated following the power rule, which yields us the antiderivative before calculating the definite integral using the Fundamental Theorem of Calculus.