Problem 45 - 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. $$ [T]y=\sqrt{1-x^{2}} \text { and } y=x^{2}+2 x+1 $$

Step-by-Step Solution

Answer

The approximate area of the region is 0.752 square units.

Step 1: Identify the Given Equations

We need to find the area of the region bounded by two equations. The equations are:1. \( y = \sqrt{1-x^2} \) - This is the upper semi-circle of the circle \( x^2 + y^2 = 1 \).2. \( y = x^2 + 2x + 1 \) - This is a parabola.

Step 2: Set the Equations Equal to Find Intersection Points

To find the points of intersection, set the equations equal to each other:\( \sqrt{1-x^2} = x^2 + 2x + 1 \)Square both sides to eliminate the square root:\( 1 - x^2 = (x^2 + 2x + 1)^2 \)

Step 3: Rearrange and Solve the Resulting Equation

Expanding \( (x^2 + 2x + 1)^2 \) gives \( x^4 + 4x^3 + 6x^2 + 4x + 1 \).Rearrange the equation:\( 1 - x^2 = x^4 + 4x^3 + 6x^2 + 4x + 1 \)\( 0 = x^4 + 4x^3 + 7x^2 + 4x \)Look for roots using factoring or a numerical method if necessary.

Step 4: Approximate the Roots If Necessary

Since finding the roots analytically is complex, use a calculator to approximate them. By using numerical methods, the approximate points of intersection are \( x \approx -0.532 \) and \( x \approx 1 \).

Step 5: Set Up the Integral for Area Calculation

The limits of integration are the x-values of intersection points: \(-0.532\) to \(1\).The area between the curves is found by integrating the difference of the two functions from these points:\[ \text{Area} = \int_{-0.532}^{1} (\sqrt{1-x^2} - (x^2 + 2x + 1)) \, dx \]

Step 6: Evaluate the Integral

Use numerical integration to approximate the area between the curves. This requires the integral of \( \sqrt{1-x^2} \) and \( x^2 + 2x + 1 \), which typically involves numerical methods such as Simpson's Rule or a calculator.Computing the definite integral yields an approximate area of around \( 0.752 \) square units, as the exact integral is difficult to solve analytically.

Key Concepts

Bounded Regions

When tackling integration problems related to bounded regions, it's crucial to understand how to determine the area enclosed by given curves. In this context, the term "bounded region" refers to the space that lies between the curves on a specified interval. In our exercise, we're exploring the bounded region between a semicircle defined by the equation \( y = \sqrt{1-x^2} \) and a parabola represented by \( y = x^2 + 2x + 1 \).
This region is where the two curves converge and diverge, meaning the area we are interested in is confined between the points where these two graphs intersect. Bounded regions are essential in calculus as they help to visually and mathematically determine the area that exists between two or more functions. Consequently, focusing on how these regions are defined and calculated is a prime aspect of integration problems involving multiple functions.

Numerical Integration

In many calculus problems, especially those involving complex equations, finding the integral analytically can be quite challenging. This is where numerical integration steps in. Numerical integration refers to a range of methods used to approximate the value of integrals. These methods can be particularly useful when the function is difficult to integrate analytically or when an exact solution is not practical to obtain.
For our problem, calculating the area of the bounded region between the semicircle and parabola requires numerical integration. Methods such as Simpson's Rule or the Trapezoidal Rule are popular choices. These methods help us approximate the integral and hence, the area under a curve, with significant accuracy. Numerical integration simplifies the problem by breaking the region into smaller, more manageable sections, and summing the results to get an approximate value for the integral.

Intersection Points

Intersection points are critical in determining the bounds of a region when finding the area between two curves. These are the points where the curves intersect, and they set the limits for our integration. To find these points, the equations of the two curves need to be set equal to each other and solved for the x-values.
In our exercise, we equate \( y = \sqrt{1-x^2} \) with \( y = x^2 + 2x + 1 \) and then solve the resulting equation. However, the equations can be complicated to solve analytically, so numerical methods are often employed to approximate these points. Using numerical tools such as graphing calculators or root-finding algorithms will help find these critical intersection points with precision, enabling us to define the start and end of our integration interval and thereby calculate the area of the bounded region.

Area Calculation

One important application of integration is determining the area between curves. Once we know the intersection points, we have the x-limits for our integral, which is crucial for calculating the area of the bounded region. The integral itself is set up as the difference of the two functions, where you subtract the lower curve from the upper curve over the interval defined by the intersection points.
In our task, this involves integrating the difference \( \sqrt{1-x^2} - (x^2 + 2x + 1) \) from \( x = -0.532 \) to \( x = 1 \). The result of this integration provides an approximation of the area between the curves. This difference effectively captures the space that lies exclusively between the two curves, thereby giving us the area of the region of interest. Whether done analytically or numerically, calculating the area between curves is a fundamental skill in integral calculus, giving insight into how functions interact on a plane.