Question

Evaluate each improper integral or show that it diverges. $$ \int_{2}^{4} \frac{d x}{\sqrt{4 x-x^{2}}} $$

Short answer

The integral converges, and its value is \( \pi \).

Step-by-step solution

Recognize the Structure

The integral is given by \( \int_{2}^{4} \frac{d x}{\sqrt{4 x-x^{2}}} \). Notice that the integrand is in the form involving \( \sqrt{4x-x^2} \). This suggests a trigonometric substitution because it resembles the form \( a^2 - x^2 \).

Rewrite the Radicand

Factor the expression inside the square root to become \( \sqrt{4x-x^2} = \sqrt{-x^2+4x} = \sqrt{-(x^2-4x)} = \sqrt{-(x-2)^2 + 4} \).

Use Trigonometric Substitution

Implement the substitution \( x-2 = 2\sin(\theta) \), leading to \( x = 2\sin(\theta) + 2 \). This implies \( dx = 2\cos(\theta) \, d\theta \). Substitute into the integral to simplify it.

Change Limits of Integration

When \( x = 2 \), \( \theta = -\frac{\pi}{2} \). When \( x = 4 \), \( \theta = \frac{\pi}{2} \). The new limits of integration are from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).

Simplify the Integral

Rewrite the integrand using substitution: \( \int_{-rac{\pi}{2}}^{\frac{\pi}{2}} \frac{2\cos(\theta) \, d\theta}{\sqrt{4(2\sin(\theta) + 2) - (2\sin(\theta) + 2)^2}} = \int_{-rac{\pi}{2}}^{\frac{\pi}{2}} d\theta \) because the denominator simplifies to \( 2\cos(\theta) \).

Evaluate the Trigonometric Integral

The integral \( \int_{-rac{\pi}{2}}^{\frac{\pi}{2}} d\theta \) is simple to evaluate. Its integral is \( \theta \), evaluated from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).

Perform Final Calculation

Evaluate \( \left[ \theta \right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}} = \frac{\pi}{2} - (-\frac{\pi}{2}) = \pi \).

State the Result

Since the integral simplifies to an exact value and does not tend to infinity, it converges. The value of the integral is \( \pi \).

Key concepts

Trigonometric Substitution

Trigonometric substitution is a powerful technique in calculus used to evaluate integrals involving square roots of quadratic expressions. In this particular problem, the expression under the square root is \( \sqrt{4x - x^2} \). This resembles the form \( a^2 - x^2 \), suggesting the use of a trigonometric identity. By factoring, you can rewrite it as \( \sqrt{-(x-2)^2 + 4} \). This reformulates the expression to fit a classic trigonometric substitution form.

In this case, using the substitution \( x - 2 = 2\sin(\theta) \) helps simplify the integral. Here, the trigonometric substitution converts the complicated square root expression into a much more manageable form. Choosing a sine function makes it easier since \( 1 - \sin^2(\theta) \) can directly be replaced by \( \cos^2(\theta) \), which simplifies deriving and evaluating the integral.

This substitution not only enables an easier integration process but often also leads to more straightforward computations in evaluating definite integrals.

Limits of Integration

When performing integration, especially with substitutions, it is essential to also transform the limits of integration from the variable \( x \) to the new variable \( \theta \). This ensures the bounds relate to the angle that corresponds to changes in the original expression.

For this problem, the original limits for \( x \) are 2 to 4. Using the substitution \( x - 2 = 2\sin(\theta) \), these are converted by substituting back into the expression. The lower limit \( x = 2 \) implies \( \theta = -\frac{\pi}{2} \), and the upper limit \( x = 4 \) implies \( \theta = \frac{\pi}{2} \).

These new limits for \( \theta \) range from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \). This transformation ensures the integration process remains consistent with respect to the substitution, covering the equivalent range for \( \theta \) after conversion.

Integral Convergence

An integral is classified as convergent if its value results in a finite number. With improper integrals, which are integrals that cover an infinite interval or have an integrand approaching infinity, particular care is needed to determine whether they converge or not.

In this example, by simplifying the integrand through trigonometric substitution, the integral becomes easy to compute, simplifying to \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} d\theta \). The fact that this results in a finite value, namely \( \pi \), indicates that the integral is convergent.

This concept of convergence is crucial in calculus, as it determines whether the integration process results in a meaningful, finite value as opposed to tending towards infinity.

Evaluating Definite Integrals

Evaluating a definite integral involves computing the integral of a function over a specific interval. Once the integral has been simplified, solving becomes a matter of applying limits and calculating the resulting expression.

In our problem, after trigonometric substitution and simplification, the integral reduces to a form that is straightforward: \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} d\theta \). The evaluation involves finding the antiderivative, \( \theta \), and applying the limits.

Plugging \( \theta \) back into the bounds yields \( \left[ \theta \right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \), which simplifies to \( \pi \). This final step is crucial in confirming that the calculated value represents the entire area under the curve, or function, within the specified limits.