Problem 315 - Calculus Volume 1

Question

Find the area under the graph of \(g(t)=\frac{t}{\left(1-t^{2}\right)^{a}}\) between \(t=0\) and \(t=x,\) where \(0 < x < 1\) and \(a > 0\) is fixed. Evaluate the limit as \(x \rightarrow 1\)

Step-by-Step Solution

Answer

The integral evaluates to a finite limit if and only if \( a > 1 \).

Step 1: Setup the Integral

To calculate the area under the curve defined by the function \( g(t) = \frac{t}{(1-t^2)^a} \) from \( t = 0 \) to \( t = x \), we set up an integral:\[A(x) = \int_{0}^{x} \frac{t}{(1-t^2)^a} \, dt\]

Step 2: Change of Variables

Consider the substitution \( u = 1 - t^2 \), which implies that \( du = -2t \, dt \). This substitution simplifies the integral. When \( t = 0, u = 1 \) and when \( t = x, u = 1 - x^2 \).After substitution, the integral becomes:\[A(x) = \int_{1}^{1-x^2} \frac{-1}{2} \cdot u^{-a} \, du = \frac{1}{2} \int_{1-x^2}^{1} u^{-a} \, du\]

Step 3: Evaluate the Definite Integral

The integral \( \int u^{-a} \, du = \frac{u^{1-a}}{1-a} \), as long as \( a eq 1 \). We evaluate the definite integral:\[\frac{1}{2(1-a)} \left[ (1)^{1-a} - (1-x^2)^{1-a} \right] = \frac{1}{2(1-a)} \left[ 1 - (1-x^2)^{1-a} \right]\]

Step 4: Evaluate the Limit as \(x \rightarrow 1\)

To find \( \lim_{x \to 1} A(x) \), evaluate:\[\lim_{x \to 1} \frac{1}{2(1-a)} \left[ 1 - (1-x^2)^{1-a} \right]\]As \( x \to 1 \), \( 1-x^2 \to 0 \). Since \( a > 0\), \((1-x^2)^{1-a} \) tends towards infinity if \( 1-a < 0 \) (or, equivalently, \( a > 1 \)), leading to a finite limit. However, if \( a \leq 1 \), the term results in a zero or undefined value, respectively.

Key Concepts

Definite Integral

The definite integral is a powerful tool used in calculus to calculate the total area under a curve over a specific interval. This process transforms the function over an interval into a single number, representing this area. For the problem at hand, we are interested in the area between the curve defined by the function \( g(t) = \frac{t}{(1-t^2)^a} \) and the x-axis, from \( t = 0 \) to \( t = x \). Here, the integral is defined as:

• \( A(x) = \int_{0}^{x} \frac{t}{(1-t^2)^a} \, dt \)

To solve for the definite integral, we begin by setting up this integral expression, which starts our journey to find the exact area enclosed. This integral evaluates how the function behaves over this specified region, crucial for getting accuracy in our calculations. Remember, the limits \( 0 \) and \( x \) are what make this integral 'definite', giving it finite boundaries to operate within.

Change of Variables

The method of change of variables, or substitution, is a technique often employed in integration to simplify complex integrals. It involves introducing a new variable to make the integration process easier. In this exercise, we employ the substitution \( u = 1 - t^2 \). This substitution is strategic because it converts the given variable from \( t \) to \( u \), simplifying the expression.

• Differentially, \( du = -2t \, dt \), letting us rearrange the integral's differential component.
• By substituting, the integration limits also transform: when \( t = 0 \), \( u = 1 \); when \( t = x \), \( u = 1 - x^2 \).
• The original integral expression becomes:
  • \( A(x) = \int_{1}^{1-x^2} \frac{-1}{2} \cdot u^{-a} \, du \)

This change reduces the problem from an integral difficult to evaluate directly into a more manageable one by simplifying the variable relation and focusing on \( u \). The result is a rewritten integral where the evaluation becomes straightforward.

Limit Evaluation

Limit evaluation is a fundamental aspect of calculus, often used to determine the behavior of a function as it approaches a particular value. In this exercise, after finding the integral, the final step is to evaluate the limit of the result as \( x \rightarrow 1 \).

• The expression obtained from evaluating the integral is:
  • \( \lim_{x \to 1} \frac{1}{2(1-a)} \left[ 1 - (1-x^2)^{1-a} \right] \)
• As \( x \to 1 \), \( 1-x^2 \to 0 \).
• Special considerations arise depending on the value of \( a \):
  • If \( a > 1 \), since \( (1-x^2)^{1-a} \rightarrow \infty \), the integral has a finite and meaningful limit.
  • If \( a \leq 1 \), complicated outcomes can occur, such as results tending towards zero or being undefined.

Understanding limits is pivotal here because it demonstrates how the area under the curve behaves as you approach the boundary set in the problem. Thus, evaluating this limit helps us realize not just the integral's final result but also its applicability under varying conditions of \( a \).