Problem 4 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(1-34\) without using tables. $$ \int_{0}^{4} \frac{d x}{\sqrt{4-x}} $$

Step-by-Step Solution

Answer

The value of the integral is 4.

Step 1: Identify the Integration Formula

Notice that the integral \( \int \frac{dx}{\sqrt{a - x}} \) resembles the integral identity used for evaluating integrals of the form \( \int \frac{dx}{\sqrt{a^2 - x^2}} \). Here we have \( a = 2 \) and \( a^2 = 4 \).

Step 2: Use Substitution

We can use the substitution \( x = 4 - u^2 \). Then, \( dx = -2u \ du \). Adjust limits as well: when \( x = 0 \), \( u = 2 \), and when \( x = 4 \), \( u = 0 \).

Step 3: Change the Integral Limits and Expression

Substituting into the integral, \( \int_{0}^{4} \frac{dx}{\sqrt{4-x}} \) becomes \( \int_{2}^{0} \frac{-2u \, du}{\sqrt{4-(4-u^2)}} \).The denominator simplifies to \( \int_{2}^{0} \frac{-2u \, du}{u} \), which results in \( \int_{2}^{0} -2 \, du \).

Step 4: Simplify and Evaluate the Integral

This can be simplified to \( 2 \int_{0}^{2} du \), as we switch the order of integration. It is equal to \( 2 [u]_{0}^{2} \), which evaluates to \( 2(2) - 2(0) = 4 \).

Step 5: Final Evaluation of the Integral

Finally, the evaluated result of \( \int_{0}^{4} \frac{dx}{\sqrt{4-x}} \) is \( 4 \).

Key Concepts

Substitution Method

The Substitution Method is a powerful technique used to simplify integrals by changing variables. It is often referred to as "u-substitution". This method can make complicated integrals more manageable. The main idea is to select a new variable, "u", to replace part of the integrand. This new variable should simplify the expression, typically turning a complex function into a simpler one.

• Identify a portion of the integrand to substitute with a new variable "u".
• Calculate the derivative of "u", known as "du".
• Replace the original variable and differential with "u" and "du".
• Adjust the limits of integration if it’s a definite integral.

In our problem, we chose the substitution \( x = 4 - u^2 \), which transforms the differential \( dx \) into \( -2u \, du\). This choice simplifies the original integral, reducing it to a form that is easy to integrate, \( \int_{2}^{0} -2 \, du \).

Definite Integrals

Definite Integrals extend the concept of an integral by including limits of integration. These limits define the range over which the function is to be integrated, making it possible to compute a specific numerical answer. Unlike indefinite integrals, which represent families of functions, definite integrals have a precise value.

• The integral \( \int_{a}^{b} f(x) \, dx \) calculates the signed area between the function \( f(x) \) and the x-axis, from \( x = a \) to \( x = b \).
• Definite integrals use the Fundamental Theorem of Calculus, which connects integration with differentiation.
• When using the substitution method, the limits must also change according to the substitution.

In the example, the integral from 0 to 4 becomes an integral from 2 to 0 after introducing the substitution. This change highlights the importance of adjusting limits when variables are transformed.

Integral Evaluation

Integral Evaluation refers to the process of finding the value of an integral, whether definite or indefinite. It involves selecting the appropriate method, simplifying the integral, and calculating the final result.

• One must recognize patterns and use algebraic transformations to ease the integration process.
• For definite integrals, determine the antiderivative and apply the limits to find the specific result.
• Simplification before solving helps make difficult integrals approachable.

In our problem, after substitution and adjustments, the integral simplifies to \( \int_{2}^{0} -2 \, du \). Evaluating this integral, we switch the limits and find \( 2 \int_{0}^{2} du \), which is straightforwardly calculated as \( 2 \times 2 = 4 \). This shows how simplification and careful calculation lead to the desired solution.

Mathematical Limits

Mathematical Limits are essential in integration, especially when dealing with definite integrals or changing variables. Understanding limits ensures the integrity of substitutions and manipulations in an integral.

• When substituting, update each limit to reflect the new variable.
• Maintain the order of integration, switching limits when necessary to match the direction of integration.
• Ensure all expressions are correctly translated under the new limits after substitution.

In the substitution step, \( x = 0 \) originally becomes \( u = 2 \), and \( x = 4 \) becomes \( u = 0 \). This reversal changes the integration direction, so limits are switched to properly evaluate the integral. Proper limit handling ensures that integral evaluation is accurate and complies with mathematical principles.