Problem 10 - Calculus

Question

Evaluate the indicated integrals. $$ \int_{2}^{3} \frac{y^{2}-1}{\left(y^{3}-3 y\right)^{2}} d y $$

Step-by-Step Solution

Answer

The integral diverges, as it is undefined at \(y=3\).

Step 1: Identify Substitution Variable

We notice that the integrand might be simplified by substitution. Let's select a substitution that simplifies the denominator. Observe that the denominator is \((y^3 - 3y)^2\). We can set \(u = y^3 - 3y\).

Step 2: Differentiate the Substitution

Differentiate the substitution \(u = y^3 - 3y\) to find \(du\) in terms of \(dy\).\[ \frac{du}{dy} = 3y^2 - 3 \]This implies that:\[ du = (3y^2 - 3) dy \]

Step 3: Simplify the Integral

Rewriting the integrand using our substitution, we substitute \(y^2 - 1\) from \(du\).Notice that \(y^2 - 1 = \frac{1}{3}(3y^2 - 3)\).Substitute into the integral to get:\[\int \frac{1}{3} \frac{du}{u^2}\]

Step 4: Evaluate the Integral

The integral \(\int \frac{du}{u^2}\) is straightforward, resulting in:\[\int -\frac{1}{u} du = -\frac{1}{u} + C\]

Step 5: Substitute Back to Original Variable

Substitute \(u = y^3 - 3y\) back into the expression:\[-\frac{1}{y^3 - 3y} + C\]

Step 6: Calculate Definite Integral

Using the evaluation limits from \(y=2\) to \(y=3\):\[ \left[-\frac{1}{(3^3 - 3 \times 3)} + \frac{1}{(2^3 - 3 \times 2)}\right] \]Calculate the values:For \(y=3\): \[-\frac{1}{0} = \text{undefined}\]For \(y=2\): \(-\frac{1}{2}\)Since \(\frac{1}{0}\) is undefined, the integral diverges.

Key Concepts

Definite Integral

A definite integral is used to calculate the area under a curve between two limits along the x-axis. Unlike indefinite integrals, which include an arbitrary constant, definite integrals provide a specific numerical value. In our exercise, we aim to evaluate the integral of a function from 2 to 3:\[\int_{2}^{3} \frac{y^{2}-1}{\left(y^{3}-3y\right)^{2}} \, dy \]The process involves several steps:

• Setup: Determining the limits of integration, in this case, 2 and 3, provides the "definite" aspect of this integral.
• Calculation: We then carry out the process of integration, ensuring we evaluate at both limits to get an exact value.
• Interpretation: The result tells us the area between the curve and the y-axis from y=2 to y=3.

In practice, a well-computed definite integral lets us accurately understand physical spaces and systems modeled by curve behavior.

Substitution Method

The Substitution Method is a powerful technique in calculus used to simplify complex integrals. This method involves choosing a new variable that will simplify the integral's form. In this exercise, we used:\[ u = y^3 - 3y \]Differentiating, we find:\[ du = (3y^2 - 3) \, dy \]Here are some steps to utilize the substitution method effectively:

• Identify the Part to Simplify: Look for expressions in the integrand that can be replaced for easier integration.
• Implement the Substitution: Rewrite the integral in terms of the new variable by expressing dy in terms of du.
• Calculate the New Integral: Once simplified, compute the integral, which is straightforward and often results in a simpler form.
• Re-substitute Original Variables: After calculating, replace back into the original variable terms.

In our exercise, substitution allowed us to transform a complicated ratio into a straightforward form for integration.

Integral Divergence

In calculus, an integral diverges when it doesn't converge to a finite value. This often indicates that the area under the curve is infinite or undefined between the given limits. This happened in our exercise because:

• Undefined Value: As we calculated the definite integral from y=2 to y=3, we reached a part of the function that gave 0 in the denominator.
• 1/0 Issue: Specifically, at y=3, the term \[-\frac{1}{(y^3 - 3y)}\] resulted in an undefined \(-\frac{1}{0}\).

When an integral results in infinite or undefined limits, we say it diverges. This outcome is crucial in understanding the function's behavior. For some physical or theoretical models, a diverging integral might suggest a point of change or anomaly in the system being analyzed.