Problem 26 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the definite integral. $$\int_{1}^{4} 2 x^{-3 / 2} d x$$

Step-by-Step Solution

Answer

The definite integral \(\int_{1}^{4} 2 x^{-3 / 2} d x\) can be evaluated as follows: Rewrite the function as \(\int_{1}^{4} \frac{2}{x^{3/2}} d x\). Then, find the antiderivative using the reverse power rule, which results in \(-4x^{-1/2}\). Finally, apply the limits of integration: \(-4(4^{-1/2}) - (-4(1^{-1/2})) = 2\). Therefore, the value of the definite integral is \(2\).

Step 1: Rewrite the function using negative exponents

Rewrite the given function so that negative exponents are written as positive exponents in the denominator. This makes it easier to integrate: \[\int_{1}^{4} 2 x^{-3 / 2} d x = \int_{1}^{4} \frac{2}{x^{3/2}} d x\]

Step 2: Find the antiderivative

Find the antiderivative of the function \(\frac{2}{x^{3/2}}\). To integrate this function, use the reverse power rule, where the antiderivative of \(x^n\) is \(\frac{x^{n+1}}{n+1}\): \[F(x) = \int \frac{2}{x^{3/2}} d x = 2\int x^{-3/2} d x = 2\frac{x^{-1/2}}{-1/2} = -4x^{-1/2}\]

Step 3: Evaluate the integral for the limits of integration

Plug in the limits of integration into the antiderivative you found. Then, subtract the lower limit value from the upper limit value to get the final answer: \[F(4) - F(1) = -4(4^{-1/2}) - (-4(1^{-1/2})) = -4(\frac{1}{2}) - (-4(1)) = -2 + 4 = 2\] The value of the definite integral is 2.

Key Concepts

Antiderivative

An antiderivative is like playing in reverse. Imagine you're retracing your steps to discover the original path. This concept is crucial in integration because it allows us to find the function that, when differentiated, gives us the original function we started with. In the exercise, we found the antiderivative of the function \( \frac{2}{x^{3/2}} \) by applying the reverse power rule.

To unpack it further:

• We begin with \( \frac{2}{x^{3/2}} \), which is reformulated as \( 2x^{-3/2} \) for ease of integration.
• The antiderivative of a function is akin to its indefinite integral. It provides a new function, \( F(x) \), that represents all possible area under the curve from a point to \( x \).
• In this step, getting to \( F(x) = -4x^{-1/2} \) indicates what the function was before it was differentiated to \( \frac{2}{x^{3/2}} \).

By finding the antiderivative, we lay the groundwork for evaluating definite integrals.

Integration

Integration is the process of finding the area under a curve on a graph. This process is central to calculus and involves summing up infinitesimally small slices under a curve to find a total area.

Our exercise focused on integrating the function \( \frac{2}{x^{3/2}} \) over a specific range of \( x \) values, from 1 to 4. Here's what you need to know:

• Integration turns a rate of change or a slope (known as a derivative) back into the original amount it was representing - hence 'anti-differentiation.'
• There are two types of integrals: indefinite, which gives a function (the antiderivative), and definite, which calculates the area between the curve and the x-axis over a given interval.
• In this task, we computed the definite integral to find the total accumulated quantity from \( x = 1 \) to \( x = 4 \). This involved determining the result of subtracting \( F(1) \) from \( F(4) \).

Integration is not just about computation but understanding the accumulation and transformation of quantities.

Reverse Power Rule

The reverse power rule is a helpful tool to unravel a function back to its roots. This method guides us in finding the antiderivative when integrating power functions. In our solution, we used the reverse power rule to integrate \( x^{-3/2} \).

Here's a breakdown:

• Normally, the derivative of \( x^n \) is \( nx^{n-1} \). The reverse process, or its antiderivative, involves doing the opposite.
• In the reverse power rule, we integrate \( x^n \) to obtain \( \frac{x^{n+1}}{n+1} \) where \( n eq -1 \). This flips the direction, allowing us to go from the rate of change (derivative) to the total quantity (integral).
• In our problem, \( n = -3/2 \), so we added 1 to get \( -1/2 \) as the new exponent, then divided by the new exponent to solve for the antiderivative.

Using the reverse power rule, we made sense of how to transform \( 2x^{-3/2} \) back to its integrated form.