Problem 12 - Calculus Concepts

Question

For Activities 5 through \(16,\) evaluate the improper integral. $$ \int_{2}^{\infty} \frac{4}{\sqrt[4]{x}} d x $$

Step-by-Step Solution

Answer

The integral diverges.

Step 1: Analyze the Properness of the Integral

The integral \( \int_{2}^{\infty} \frac{4}{\sqrt[4]{x}}\, dx \) is an improper integral because the upper limit of integration is infinity. We'll need to evaluate it as a limit.

Step 2: Rewrite the Integral with a Limit

Rewrite the integral as a limit:\[ \lim_{b \to \infty} \int_{2}^{b} \frac{4}{\sqrt[4]{x}} \, dx \]

Step 3: Simplify the Integrand

Rewrite \( \frac{4}{\sqrt[4]{x}} \) as \( 4x^{-1/4} \). The expression is now easier to integrate.

Step 4: Find the Antiderivative

Find the antiderivative of \( 4x^{-1/4} \). Using the power rule for integration, \( \int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C \), we get:\[ \int 4x^{-1/4} \, dx = \frac{4}{-1/4 + 1}x^{3/4} = \frac{4}{3/4}x^{3/4} = \frac{16}{3}x^{3/4} + C \]

Step 5: Evaluate the Definite Integral at the Bounds

Evaluate \( \frac{16}{3}x^{3/4} \) from \(2\) to \(b\):\[ \lim_{b \to \infty} \left(\frac{16}{3}b^{3/4} - \frac{16}{3}(2)^{3/4} \right) \]

Step 6: Take the Limit as b Approaches Infinity

Evaluate the limit:\[ \lim_{b \to \infty} \frac{16}{3}b^{3/4} - \frac{16}{3} \cdot 2^{3/4} \]Since \( b^{3/4} \) grows without bound as \( b \to \infty \), the term \( \frac{16}{3}b^{3/4} \) approaches infinity, making the whole integral diverge.

Key Concepts

Infinite Limits

In the context of integration, an improper integral often involves infinite limits, making the process a bit different from regular definite integrals. Here, we need to assess the behavior of the integral as one of its limits approaches infinity. When evaluating an integral with an upper limit of infinity, it transforms into a limit problem. For example, if we look at the integral \[ \int_{2}^{\infty} \frac{4}{\sqrt[4]{x}} \, dx \], this becomes \[ \lim_{b \to \infty} \int_{2}^{b} \frac{4}{\sqrt[4]{x}} \, dx \].
Thinking about the integral as a limit allows us to evaluate whether this value converges to a finite number or diverges to infinity. This concept is fundamental because it helps in understanding the behavior of functions over endless intervals. Recognizing when an integral has infinite limits is key. It also shows whether the area under a curve over an infinite range results in a finite number or extends indefinitely. This ensures we consider the full behavior of the function as it extends into infinity.

Power Rule for Integration

The power rule for integration is one of the simplest yet powerful techniques for finding antiderivatives. It states that for any real number \(n eq -1 \), the integral of \(x^{n}\) is \[ \int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C \]. This rule allows us to handle nearly any polynomial integrands.
For the integrand \(4x^{-1/4}\), we apply the power rule:- Increase the exponent by 1: \(-1/4 + 1 = 3/4\)- Divide by new exponent: \(\frac{4}{3/4} = \frac{16}{3}\).Thus, the antiderivative of \( 4x^{-1/4} \) is \[ \frac{16}{3}x^{3/4} + C \]. This step is crucial for solving any integral, as finding the correct antiderivative allows us to evaluate the definite integral. It’s essential to remember that after increasing the exponent by 1, the coefficient of the new term adjusts by dividing with that updated exponent. This transforms the equation into its antiderivative form, ready for actual integration over the specified limits.

Integrand Simplification

Integrand simplification is a vital part of solving integrals, especially before applying integration rules. Simplifying an expression often makes the integration process easier and more straightforward. In our example, the integrand \( \frac{4}{\sqrt[4]{x}} \)looks complicated initially. By rewriting it using properties of exponents, we get \(4x^{-1/4}\),which is much simpler to handle.
This form allows us to directly apply the power rule without needing additional steps. Simplification is particularly important because it makes spotting patterns or familiar formulas straightforward. By reducing complex fractions or roots into powers, we can avoid unnecessary complications and use basic integration rules effectively. Overall, this leads to appreciable efficiency and accuracy in the evaluation process.