Problem 9 - Basic Technical Mathematics with Calculus

Question

Find the areas bounded by the indicated curves. $$y=3 / x^{2}, y=0, x=2, x=3$$

Step-by-Step Solution

Answer

The area bounded by the curves is \(\frac{1}{2}\).

Step 1: Understand the Problem

To find the area bounded by the curves, we need to identify the region enclosed by the function \(y = \frac{3}{x^2}\), the x-axis \(y = 0\), and the vertical lines \(x = 2\) and \(x = 3\).

Step 2: Set Up the Integral

Since the area is bounded between \(x = 2\) and \(x = 3\) along the x-axis, we will set up the definite integral of \(y = \frac{3}{x^2}\) from \(x = 2\) to \(x = 3\): \[\text{Area} = \int_{2}^{3} \frac{3}{x^2} \, dx\]

Step 3: Compute the Antiderivative

To solve the integral, we first find the antiderivative of \(\frac{3}{x^2}\). Recall that \(\frac{1}{x^2} = x^{-2}\), so:\[ \int \frac{3}{x^2} \, dx = \int 3x^{-2} \, dx = 3 \cdot \left(-\frac{1}{x}\right) = -\frac{3}{x}\]

Step 4: Evaluate the Definite Integral

Now, evaluate the antiderivative from \(x = 2\) to \(x = 3\): \[\text{Area} = \left[ -\frac{3}{x} \right]_{2}^{3} = \left(-\frac{3}{3}\right) - \left(-\frac{3}{2}\right)\]Simplify to:\[= -1 + \frac{3}{2} = \frac{1}{2}\]

Step 5: Conclusion

The calculated definite integral represents the area of the region bounded by the curves. The area is \(\boxed{\frac{1}{2}}\).

Key Concepts

Area Under a Curve

Understanding the area under a curve is vital when solving problems involving definite integrals. In calculus, "area under a curve" typically refers to the area between the curve of a function and the x-axis within specific bounds. In this manner, if we take a look at the function given, \(y = \frac{3}{x^2}\), the problem asked us to calculate the area between this curve and the x-axis, specifically between \(x = 2\) and \(x = 3\).

• The essence of finding the area under a curve is using definite integrals, as this process replaces the concept of summing infinitely many infinitesimally thin rectangles under the curve.

This approach helps us understand real-world applications, such as measuring distances, calculating profits or growth, and evaluating physical phenomena.

Antiderivative

An antiderivative, also known as an indefinite integral, is a function that reverses the process of differentiation. When faced with the integral \( \int \frac{3}{x^2} \, dx \), we're asked to find a function for which the derivative returns the original function, \( \frac{3}{x^2} \).

• For the function \( \frac{1}{x^2} = x^{-2} \), the standard power rule for integration tells us we need to adjust the exponent.
• Thus, \( \int 3x^{-2} \, dx \) results in \( -\frac{3}{x} \), which is the antiderivative we use for further calculations.

Recognizing and finding antiderivatives are key skills necessary for computing definite integrals.

Integration Techniques

There are various techniques applied in the realm of integration. Each technique caters to different types of functions and results in finding areas or solving differential equations. For the function \( \frac{3}{x^2} \), the chosen technique was applying the power rule for integration, a straightforward and commonly used method.

• The power rule transforms \( x^{-2} \) into \( -\frac{1}{x} \), which enables us to solve the definite integral.
• This involves calculating the difference between the evaluated antiderivative at the upper and lower bounds, \( x = 3 \) and \( x = 2 \) in this context.

This method is foundational but efficient, especially for simple polynomial or rational functions. Other techniques might include substitution, integration by parts, or partial fraction decomposition, but the chosen method reflects the function's simplicity and the bounds provided.