Problem 50 - Calculus Early Transcendentals

Question

Evaluate the following definite $$\int_{1}^{\sqrt{2}} \frac{d x}{x^{2} \sqrt{4-x^{2}}}$$

Step-by-Step Solution

Answer

To evaluate the given definite integral, we followed these steps: 1. We made an appropriate substitution u = x^2 to simplify the integrand. 2. We rewrote the integral in terms of the new variable u. 3. We evaluated the transformed integral using another substitution (v = 4u-u^2) and then computed the integral in terms of v. 4. We transformed the result back to the original variable x. The final answer for the definite integral is: -(\sqrt{4} - \sqrt{3}).

Step 1: Identifying an appropriate substitution

In order to simplify the expression, we can make a substitution for the variable inside the square root. Let's use the substitution: $$u = x^2$$ With this substitution, we get: $$\frac{d u}{d x} = 2x$$ And we can express the differential du as: $$d u = 2x\,d x$$ Now, let's change the limits of integration according to the substitution: $$x = 1 \Rightarrow u = 1^2 = 1$$ $$x = \sqrt{2} \Rightarrow u = (\sqrt{2})^2 = 2$$

Step 2: Rewrite the integral in terms of u and dx

We can now rewrite the integral in terms of the new variable u: $$\int_{1}^{\sqrt{2}} \frac{dx}{x^{2} \sqrt{4-x^{2}}} = \int_{1}^{2} \frac{1}{(u) \sqrt{4-u}} \frac{du}{2x}$$ We know that \(u = x^2\), so \(x = \sqrt{u}\). Substituting this back into the integral, we get: $$\int_{1}^{2} \frac{1}{(u) \sqrt{4-u}} \frac{du}{2\sqrt{u}}$$

Step 3: Evaluating the transformed integral

We can now evaluate the integral: $$\frac{1}{2}\int_{1}^{2} \frac{du}{u \sqrt{u (4-u)}}$$ Here, the u's in the numerator and denominator cancel out, giving: $$\frac{1}{2}\int_{1}^{2} \frac{du}{\sqrt{4u-u^2}}$$ Let's make another substitution to simplify the integrand further: $$v = 4u-u^2$$ From this, we have that: $$\frac{d v}{d u} = 4-2u$$ Or: $$d v = (4-2u)\,d u$$ For our new integration limits: $$u = 1 \Rightarrow v = 4-1^2 = 3$$ $$u = 2 \Rightarrow v = 4-2^2 = 0$$ Our integral now becomes: $$\frac{1}{2}\int_{3}^{0} \frac{d v}{\sqrt{v}}$$ The upper and lower limits are switched, which means we can change the sign of the integral: $$-\frac{1}{2}\int_{0}^{3} \frac{d v}{\sqrt{v}}$$

Step 4: Evaluate the remaining integral and transform back to x

First, we evaluate the remaining integral in terms of v: $$-\frac{1}{2}\int_{0}^{3} \frac{d v}{\sqrt{v}} = -\frac{1}{2} [2\sqrt{v}\big|^3_{0}$$ Now substituting the result back to u: $$= -\big[\sqrt{4u-u^2}\big|^2_{1}$$ And finally, back to x: $$= -\big[\sqrt{4x^2-x^4}\big|^{\sqrt{2}}_{1}$$ After evaluating the limits, we have the final answer: $$= -(\sqrt{4(\sqrt{2})^2-(\sqrt{2})^4}-\sqrt{4(1)^2-(1)^4}) = -(\sqrt{4} - \sqrt{3})$$

Key Concepts

Substitution Method

The substitution method is a powerful tool in definite integration, especially when dealing with complex integrals that involve expressions like square roots or trigonometric functions. The main idea is to simplify the integrand by substituting a part of it with a new variable.
In the given exercise, the substitution is performed using:

• Substitute: \( u = x^2 \)
• Derivative: \( \frac{d u}{d x} = 2x \)
• Differential: \( d u = 2x \, d x \)

Through this substitution, we change the variable of integration from \( x \) to \( u \). This simplifies the integral and often reveals an easier path to its evaluation. Changing variables is analogous to translating a complex sentence into an easier, more understandable one.

Limits of Integration

Calculating definite integrals requires exact limits of integration. These limits are the boundary or starting and ending points of the integral on the axis it's being evaluated on. When you use substitution, the limits of integration must be adjusted according to the new variable.
During the exercise:

• Original limits: \( x = 1 \) to \( x = \sqrt{2} \)
• Transformed limits: \( u = 1 \) to \( u = 2 \)

This translation ensures the integral bounds are correctly mapped onto the new variable. It helps avoid miscalculations and ensures the area calculated under the curve remains consistent with the original problem.

Integrand Transformation

Integrand transformation involves rewriting the function inside the integral after a substitution has been made. This step is crucial for simplifying the function into a form that's easier to tackle mathematically.
For example, in the problem:

• Initial transformation: \( \int_{1}^{\sqrt{2}} \frac{dx}{x^{2} \sqrt{4-x^{2}}} \)
• After first substitution: \( \int_{1}^{2} \frac{1}{(u) \sqrt{4-u}} \frac{du}{2\sqrt{u}} \)
• Further transformation: \( \frac{1}{2}\int_{1}^{2} \frac{du}{\sqrt{4u-u^2}} \)

These changes involve operations like dividing by the derivative constant or adjusting for cancellations within expressions. They make the original complex integrand approachable in an integration-ready form.

Calculus

Calculus is the mathematical study of continuous change. It provides tools for finding rates of change and areas under curves, which are essential in many scientific and engineering fields.
Definite integration, a concept from calculus, involves computing the area under a curve between two points. In practice, calculus allows you to evaluate how variable quantities relate through differentiation and integration.

• Differentiation: Finding the rate of change or slope of a function.
• Integration: Calculating the cumulative quantity, such as area or total accumulation.

The exercise leverages calculus to not only transform and simplify integrals via substitution but also integrate by parts of defined limits, illustrating one of its many integral solving techniques.