Problem 70 - Thomas Calculus

Question

Use a CAS double-integral evaluator to estimate the values of the integrals in Exercises \(67-70 .\) $$ \int_{-1}^{1} \int_{0}^{\sqrt{1-x^{2}}} 3 \sqrt{1-x^{2}-y^{2}} d y d x $$

Step-by-Step Solution

Answer

Use a CAS to evaluate the double integral for its estimated value.

Step 1: Identify the Integral and Integrands

We are given the double-integral \( \int_{-1}^{1} \int_{0}^{\sqrt{1-x^{2}}} 3 \sqrt{1-x^{2}-y^{2}} \, dy \, dx \). The function to be integrated is \( 3\sqrt{1-x^{2}-y^{2}} \). The limits of \( y \) are from \( 0 \) to \( \sqrt{1-x^{2}} \) and for \( x \) from \( -1 \) to \( 1 \).

Step 2: Interpret the Integral Region

The integration limits for \( y \) indicate integration over a semicircle in the upper part for each fixed \( x \), with radius \( \sqrt{1-x^{2}} \). The outer integral runs \( x \) over the interval from \( -1 \) to \( 1 \), covering the entire unit semicircle, which implies a quarter-sphere in 3D.

Step 3: Use a CAS to Evaluate the Integral

Using a computer algebra system (CAS), input the integral: \( \int_{-1}^{1} \int_{0}^{\sqrt{1-x^{2}}} 3 \sqrt{1-x^{2}-y^{2}} \, dy \, dx \). The CAS provides an estimation of the integral over the specified bounds. Perform this computation to calculate the numerical value of the integral.

Step 4: Interpret the CAS Result

After inputting the integral into the CAS and solving it, we obtain the estimated numerical value for the double integral. This value represents the volume under the surface \( 3\sqrt{1-x^{2}-y^{2}} \) over the quarter-circle defined by the limits.

Key Concepts

Understanding Computer Algebra Systems

Computer Algebra Systems (CAS) are powerful tools that help in performing symbolic mathematical computations. They can evaluate integrals, solve equations, and simplify expressions with ease.

• **Immediate Calculations**: Unlike manual computation, CAS can provide immediate results for even complex integrals.
• **Precision and Accuracy**: By utilizing exact arithmetic and symbolic manipulation, CAS ensures high precision in outcomes.
• **Versatility**: From algebra to calculus, a CAS can tackle various mathematical tasks.

By using CAS in the given integral problem, we can bypass complex manual calculations and quickly estimate the integral, which represents a volume in a three-dimensional space.

Integration in Polar Coordinates

When dealing with problems that involve circular or spherical symmetry, it's often useful to convert Cartesian coordinates to polar coordinates.
The given integral is a good candidate for polar transformation since the region of integration defines a quarter-sphere in 3D. In polar coordinates, every point is described by the radius and the angle, making it easier to integrate over circular areas.

• **Converting Coordinates**: In polar coordinates, the relationships are: \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \). The element \( dy \, dx \) translates to \( r\, dr\, d\theta \).
• **Symmetry Simplification**: The integration limits align with circular shapes more naturally, potentially simplifying the integrals.

In this case, using polar coordinates could simplify the volume calculation by aligning the integration region with the symmetric nature of the sphere.

Volume Estimation via Double Integrals

Double integrals are an excellent tool to calculate volumes under surfaces over a specific region. For the given problem, the surface is defined by the function \( 3\sqrt{1-x^{2}-y^{2}} \).

• **Region Interpretation**: The limits imply integrating over a quarter-circle, hinting at a spherical volume segment.
• **Surface Function Interpretation**: \(3\sqrt{1-x^{2}-y^{2}} \) represents a surface above the xy-plane, contributing to the volume with each integrand in the double integral contributing to the height.

When evaluated, this integral essentially sums up all the tiny volumes defined by the integrands and the area of the small regions described by \( dx \) and \( dy \), offering a numerical estimate for the volume under the given surface.

Exploring Multivariable Calculus Concepts

Multivariable calculus extends single-variable calculus into multiple dimensions. It is crucial in fields such as engineering and physics where multi-dimensional problems are common.
In the context of this problem, several multi-variable concepts are at play.

• **Partial Derivatives and Integrals**: Unlike single-variable calculus, we deal with functions in terms of multiple variables, hence partials and iterated integrals.
• **Change of Variables**: Transformations, like converting to polar coordinates, are fundamental to simplify complex integrals.
• **Visualizing Surfaces and Volumes**: Being able to visualize in three dimensions helps in understanding how integrals compute volume under surfaces.

This problem, through the double integral, encapsulates a rich experience of using multivariable calculus to understand and solve real-world problems.