Problem 51 - Calculus an Applied Approach

Question

Use a symbolic integration utility to evaluate the double integral. $$ \int_{0}^{1} \int_{x}^{1} \sqrt{1-x^{2}} d y d x $$

Step-by-Step Solution

Answer

The value of the double integral is \( \frac{2}{3} \)

Step 1: Understanding Double Integrals

Double integral represents a volume below the surface of integration and above xy plane. To evaluate a double integral, it is processed as an iteration of two single integrals.

Step 2: Find the Inner Integral

Keeping the outer variable 'x' constant, we start by evaluating the inner integral first. Here, the inner integral is: \( \int_{x}^{1} \sqrt{1-x^{2}} dy \). The result of the inner integration with respect to 'y' is \( y\sqrt{1-x^{2}} \) evaluated from 'x' to '1'. Thus, the inner integral is \( \sqrt{1-x^{2}} - x\sqrt{1-x^{2}} \) which simplifies to \( (1-x)\sqrt{1-x^{2}} \)

Step 3: Evaluate Outer Integral

Next, take the result of the inner integral and integrate it with respect to 'x' from '0' to '1'. Thus, the outer integral is: \( \int_{0}^{1} (1-x)\sqrt{1-x^{2}} dx \). This can be solved by making the substitution \( x=sin(u) \), which leads to the solution \( \frac{2}{3} \)

Key Concepts

Symbolic Integration

Symbolic integration is a crucial technique used in calculus to find the antiderivative or integral of a function in terms of symbols rather than numbers. In the context of double integrals, symbolic integration simplifies the process significantly, especially when dealing with complex functions.

This allows us to express the area or volume under curves and surfaces analytically. In our original problem, symbolic integration helps in breaking down the two-dimensional integral into manageable parts.

• We start by integrating with respect to one variable while keeping the other variable constant.
• This involves using symbolic expression and manipulation to arrive at an integral form that is easier to evaluate or simplify.

By using a symbolic integration utility, we can accurately evaluate expressions that might be cumbersome or even impossible to solve analytically. This process is particularly useful in mathematical problems involving functions that lack straightforward numerical solutions.

Integration by Substitution

Integration by substitution is a powerful technique often used to simplify integrals. This approach is particularly helpful when direct integration is difficult or when the integral involves compositions of functions. In the given problem, the outer integral was evaluated using substitution.

Typically, the process includes the following steps:

• Choose a substitution that simplifies the integrand. For example, using the trigonometric substitution \( x = \sin(u) \).
• Substitute both the function and differential, converting the variable of integration.
• Integrate the transformed function.
• Convert back to the original variable by reversing the substitution.

By substituting \( x = \sin(u) \), the original complex expression is transformed into a simpler one, making it easier to integrate. This substitution not only simplifies the integrand but also optimally aligns with trigonometric identities, streamlining the integration process considerably. This results in precise solutions that might be too complex to handle via elementary methods.

Iterated Integrals

Iterated integrals involve evaluating a double integral by "iterating" the process of integration over each variable sequentially. The concept is fundamental in multivariable calculus, typically used when integrating functions of two or more variables over a specific domain.

Here's how it works:

• First, you integrate with respect to the variable of the inner integral while treating the outer variable as a constant.
• Once the inner integral is solved, the expression obtained becomes the integrand of the outer integral.
• Then, you perform the outer integration over its specified limits.

In the given exercise, we start by solving the inner integral \( \int_{x}^{1} \sqrt{1-x^{2}} \, dy \). The result becomes the integrand for the outer integral, which is then solved over its limits \( x = 0 \) to \( x = 1 \). Iterated integrals allow us to systematically handle multivariable problems by breaking them down into simpler one-dimensional problems, which are easier to solve sequentially.