Problem 39 - Calculus Single Variable

Question

Sketch the given region. \(\left\\{(x, y): x^{2}+y^{2}>16\right\\}\)

Step-by-Step Solution

Answer

The region is outside (and excluding) the circle centered at the origin with radius 4.

Step 1: Identify the Definition of the Region

The inequality given is \(x^2 + y^2 > 16\). This represents a region outside a circle in the coordinate plane. The equation \(x^2 + y^2 = 16\) describes a circle centered at the origin with a radius of 4.

Step 2: Determine the Circle's Radius and Center

The circle defined by \(x^2 + y^2 = 16\) is centered at (0,0) with a radius of 4. We know this because the standard form \(x^2 + y^2 = r^2\) describes a circle with radius \(r\) and center (0,0). Since \(r^2 = 16\), the radius \(r = 4\).

Step 3: Understand the Inequality

The inequality \(x^2 + y^2 > 16\) signifies that the set of points \((x, y)\) lies outside the circle. Points exactly on the circle satisfy \(x^2 + y^2 = 16\), whereas points outside the circle satisfy \(x^2 + y^2 > 16\).

Step 4: Sketch the Circle

Sketch a circle with a center at (0,0) and with a radius of 4. Use your compass or freehand, making sure the circle passes through the points (4,0), (0,4), (-4,0), and (0,-4). Label each key point on the circle to ensure accuracy.

Step 5: Shade the Outside Region

Shade the entire region outside the circle to represent all points that satisfy \(x^2 + y^2 > 16\). Do not shade the circle or the area inside the circle because the inequality is strict (\(>\), not \(\geq\)), meaning points on the circle do not satisfy the inequality.

Key Concepts

Circle Equations

A circle is a fundamental shape in coordinate geometry. To describe a circle on a coordinate plane, you use the equation \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle, and the center is \((0,0)\). This is known as the standard form of a circle equation.
In this form, each

• \(x\) and \(y\) are variables representing any point on the circle.
• \(r^2\) is the square of the radius, which tells how far the circle extends from its center.

For example, if \(x^2 + y^2 = 16\), then the radius \(r\) is 4, since \(r^2 = 16\). This means the circle passes through the points where \(x\) or \(y\) equals 4 or -4, like (4,0), (0,4), (-4,0), and (0,-4).
This equation helps us easily understand the size and position of circles in coordinate geometry.

Sketching Regions

Sketching regions based on inequalities in the coordinate plane means identifying the set of points that satisfy the given condition or inequality. You start with the equality, which forms a boundary. Then interpret the inequality to know which side of the boundary contains points that satisfy it.
When you have an inequality like \(x^2 + y^2 > 16\), you first consider the equation \(x^2 + y^2 = 16\), which is the boundary. This equation forms a circle, and our job is to determine where the region described by the inequality lies.

• Points on the circle satisfy the equation \(x^2 + y^2 = 16\).
• Points inside the circle satisfy \(x^2 + y^2 < 16\).
• For \(x^2 + y^2 > 16\), the points are outside this circle.

To sketch such regions, draw the circle first with the correct radius and center. In this case, the radius is 4, centered at the origin.
Then, you shade the area outside this circle. Be sure not to shade the circle itself or its inside, respecting the strict inequality which excludes the boundary.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to study geometric figures using a coordinate system. This powerful tool helps in proving various geometric theorems and solving geometric problems.
In coordinate geometry, each point in the plane is identified by an ordered pair \((x, y)\). This ability to pinpoint positions through coordinates makes it easier to handle complex shapes like circles, ellipses, and parabolas.

• Circle equations show how distances and positions relate algebraically and geometrically.
• Inequalities help in defining regions and understanding parts of the plane that satisfy specific criteria.
• The use of coordinates allows us to solve real-world problems involving positioning, navigation, and design.

By converting geometric shapes into algebraic expressions and inequalities, coordinate geometry turns visual problems into mathematical ones.
This approach is particularly handy when sketching regions on a coordinate plane, as it lets us use equations and inequalities efficiently to determine which areas of the plane are included in a given set.