Problem 41 - 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. $$ [T]x=\sqrt{4-y^{2}} \text { and } y^{2}=1+x^{2} $$

Step-by-Step Solution

Answer

The area is the integral of the difference between the two curves over the interval determined by the intersection points.

Step 1: Understanding the Problem

We are asked to find the area of the region bounded by the two curves: \( x = \sqrt{4-y^{2}} \) and \( y^{2} = 1 + x^{2} \). Initially, we need to understand the curves and their possible intersections to determine the bounded region.

Step 2: Rewrite Equations in Standard Form

The first equation \( x = \sqrt{4-y^{2}} \) can be rewritten as \( x^2 + y^2 = 4 \), which represents a circle centered at the origin with a radius of 2. The second equation \( y^{2} = 1 + x^{2} \) can be rearranged to \( x^2 - y^2 = -1 \), representing a hyperbola.

Step 3: Find Intersection Points

By equating the expressions for \( y^2 \) from both equations, we get \( 4 - x^2 = 1 + x^2 \). Solving for \( x \), we find \( x^2 = \frac{3}{2} \), so \( x = \pm \sqrt{\frac{3}{2}} \). Substituting back, we find the corresponding \( y \) values.

Step 4: Use Numerical Tools for Exact Areas

These intersections \((\sqrt{\frac{3}{2}}, \frac{1}{\sqrt{2}})\) and \((-\sqrt{\frac{3}{2}}, -\frac{1}{\sqrt{2}})\) form the boundaries of the shadowed region. Calculate the area using numerical integration between these points for \( y = \sqrt{4-x^2} \) and \( y = -\sqrt{x^2+1} \).

Step 5: Calculate the Area

The definite integral needs to be calculated numerically for \( \int_{x_1}^{x_2} \left( \sqrt{4-x^2} - \left(-\sqrt{x^2+1}\right) \right) \, dx \), where \( x_1 = -\sqrt{\frac{3}{2}} \) and \( x_2 = \sqrt{\frac{3}{2}} \). Integrate to find the area.

Key Concepts

Area between curves

The area between curves is an important concept in calculus. It involves finding the space between two or more curves on a graph. To calculate the area between two curves, one typically subtracts the lower function from the upper function within a specified interval.
For this exercise, we need to find the area of the space enclosed between the circle represented by the equation \( x^2 + y^2 = 4 \) and the hyperbola \( x^2 - y^2 = -1 \). The area calculation involves integrating over the interval determined by their intersection points. This ensures that we measure the vertical distance between the two curves accurately, from left to right.
Understanding the shapes of these curves on a graph helps in visualizing the area that needs to be calculated, making the problem-solving process more intuitive.

Intersection points

Intersection points occur where two curves meet. They’re crucial for determining the limits of integration when calculating the area between curves. To find these points analytically, equate the expressions derived from the curves equations.
In this exercise, equating \( y^2 \) from each equation gave us \( 4 - x^2 = 1 + x^2 \), which simplifies to \( x^2 = \frac{3}{2} \), meaning \( x = \pm \sqrt{\frac{3}{2}} \). Substituting back to find the respective \( y \)-values results in the intersection points \( (\sqrt{\frac{3}{2}}, \frac{1}{\sqrt{2}}) \) and \( (-\sqrt{\frac{3}{2}}, -\frac{1}{\sqrt{2}}) \).
These points act as boundaries for the definite integral, ensuring we measure the area precisely within the region where the curves intersect.

Definite integral

The definite integral is a calculus tool used to find the exact area under a curve between two points. It provides a way to calculate precise measurements by accumulating an infinite number of infinitesimal quantities. For areas between curves, it's the integration of the top curve minus the bottom curve.
In our context, the definite integral calculates the area between the circle and hyperbola over the interval defined by the intersection points. The integral to compute is \( \int_{-\sqrt{\frac{3}{2}}}^{\sqrt{\frac{3}{2}}} \left( \sqrt{4-x^2} - \left(-\sqrt{x^2+1}\right) \right) \, dx \). It evaluates the total "height" between these two curves from the leftmost to rightmost intersection, giving us the desired area.
Understanding how to set up and evaluate a definite integral is key to solving problems involving areas between curves effectively.

Numerical integration

Numerical integration is a technique used when it's difficult or impossible to find an analytical solution to an integral. This method approximates the area under a curve by summing the areas of shapes that approximate the curve. Common methods include the Trapezoidal Rule and Simpson’s Rule.
In this exercise, the integration limits defined by the intersection points \( x = -\sqrt{\frac{3}{2}} \) and \( x = \sqrt{\frac{3}{2}} \) require numerical integration. Despite being able to analytically describe the functions \( \sqrt{4 - x^2} \) and \(-\sqrt{x^2+1} \), calculating their integral in closed form is tricky.
By using numerical integration methods, you can achieve an approximation of the area enclosed by these curves with high accuracy, even when classical methods fall short. This approach is especially useful in cases where the functions are complex or do not have straightforward antiderivatives.