Problem 39 - Calculus Volume 1

Question

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region. $$ [T]y=x^{2} \text { and } y=\sqrt{1-x^{2}} $$

Step-by-Step Solution

Answer

The approximate area is found by calculating the integral numerically due to the complexity of the polynomial.

Step 1: Set Equations Equal

We need to find the intersection points of the curves \( y = x^2 \) and \( y = \sqrt{1-x^2} \). To do this, set the equations equal: \( x^2 = \sqrt{1-x^2} \).

Step 2: Solve for Intersection Points

Square both sides of the equation to eliminate the square root: \( x^4 = 1 - x^2 \). Rearrange this into a polynomial: \( x^4 + x^2 - 1 = 0 \). Since this is a difficult polynomial to solve analytically, we will use a calculator.

Step 3: Use Calculator for Approximate Intersection Points

Using a calculator, solve the equation \( x^4 + x^2 - 1 = 0 \) to find approximate intersection points. The solutions are \( x \approx -0.786 \) and \( x \approx 0.786 \).

Step 4: Set Up the Integral for the Area

The area between the curves can be found by integrating the difference of the functions from one intersection point to the other. The integral is: \( \int_{-0.786}^{0.786} (\sqrt{1-x^2} - x^2) \, dx \).

Step 5: Calculate the Integral

This integral can be computed numerically using a calculator. Calculate \( \int_{-0.786}^{0.786} (\sqrt{1-x^2} - x^2) \, dx \) to obtain the approximate area.

Key Concepts

Intersection points

When determining the area between two curves, the first step is to find their intersection points. Intersection points are where the two functions have the same value for both x and y. In this exercise, we're given the equations \( y = x^2 \) and \( y = \sqrt{1-x^2} \). To find where these two meet, set them equal to each other:

• \( x^2 = \sqrt{1-x^2} \).

To facilitate solving this, eliminate the square root by squaring both sides, leading to a new equation \( x^4 + x^2 - 1 = 0 \). This is a polynomial equation that may be challenging to solve analytically. By using this squared equation, we've determined the mathematical condition under which the original graphs intersect.
To find the solutions or intersection points, often a calculator is your best friend, especially when the equations become complex. Here, the intersection points were approximated as \( x \approx -0.786 \) and \( x \approx 0.786 \). Knowing these points is critical as they define the limits of integration for calculating the area between the curves.

Polynomial equation

A polynomial equation is an expression made up of variables raised to whole number powers and their coefficients. For our purposes, we've transformed the original condition \( x^2 = \sqrt{1-x^2} \) into the polynomial \( x^4 + x^2 - 1 = 0 \). This form emerged after squaring both sides to remove the square root.

While simple polynomials can be solved using techniques such as factoring or the quadratic formula, more complex polynomials like this one often require numerical methods or a calculator. The challenge lies in finding all real roots, as these contribute to identifying the critical intersections we discussed earlier.
In this context, because the equation \( x^4 + x^2 - 1 = 0 \) does not lend itself to easy analytical solutions, we rely on technological methods to determine approximate solutions approximately: \( x \approx -0.786 \) and \( x \approx 0.786 \). The solutions you find define the regions over which you'll calculate the area between curves.

Numerical integration

Numerical integration is a technique used to approximate the value of an integral when a function is too complex to integrate analytically. In our example, after establishing the intersection points, we set up the integral \( \int_{-0.786}^{0.786} (\sqrt{1-x^2} - x^2) \, dx \) to find the area between the curves.

Numerical methods, such as the trapezoidal rule, Simpson's rule, or employing a calculator, are commonly used here. These methods estimate the value of the integral based on evaluating the function at certain points and summing the contributions over the interval.

• For instance, a calculator provides quick and accurate results for \( \int_{-0.786}^{0.786} (\sqrt{1-x^2} - x^2) \, dx \), resulting in the approximate area of the region.
• Numerical integration is essential, especially when dealing with non-standard or intricate functions.

By understanding numerical integration, we can effectively estimate the area even when the actual solution is elusive or overly complicated.