Problem 63 - Calculus and its applications

Question

Find the area of the region bounded by the given graphs. $$ x+2 y=2, y-x=1,2 x+y=7 $$

Step-by-Step Solution

Answer

The area of the region bounded by the lines is 6 square units.

Step 1: Convert Equations to Slope-Intercept Form

To identify the region bounded by the three equations, we first convert each to slope-intercept form, which is \(y = mx + b\). 1. For the equation \(x + 2y = 2\): - Solve for \(y\): \[2y = -x + 2 \Rightarrow y = -\frac{1}{2}x + 1\]2. For the equation \(y - x = 1\): - Solve for \(y\): \[y = x + 1\]3. For the equation \(2x + y = 7\): - Solve for \(y\): \[y = -2x + 7\]

Step 2: Find Intersection Points

Calculate where these lines intersect by solving pairs of equations simultaneously:1. Intersection of \(y = -\frac{1}{2}x + 1\) and \(y = x + 1\): - Set the equations equal: \[-\frac{1}{2}x + 1 = x + 1\] - Solve for \(x\): \[-\frac{1}{2}x = x \Rightarrow \frac{1}{2}x = 0 \Rightarrow x = 0\] - Substitute in \(y = x + 1\): \[y = 0 + 1 = 1\] - Intersection point: \((0, 1)\)2. Intersection of \(y = -\frac{1}{2}x + 1\) and \(y = -2x + 7\): - Set the equations equal: \[-\frac{1}{2}x + 1 = -2x + 7\] - Solve for \(x\): \[\frac{3}{2}x = 6 \Rightarrow x = 4\] - Substitute in \(y = -\frac{1}{2}x + 1\): \[y = -\frac{1}{2}(4) + 1 = -2 + 1 = -1\] - Intersection point: \((4, -1)\)3. Intersection of \(y = x + 1\) and \(y = -2x + 7\): - Set the equations equal: \[x + 1 = -2x + 7\] - Solve for \(x\): \[3x = 6 \Rightarrow x = 2\] - Substitute in \(y = x + 1\): \[y = 2 + 1 = 3\] - Intersection point: \((2, 3)\)

Step 3: Use Points to Calculate the Area by the Shoelace Formula

With the intersection points \((0, 1), (4, -1), (2, 3)\), use the shoelace formula to calculate the area of the triangle formed by these points:The shoelace formula is given by:\[ \text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_1 - (y_1x_2 + y_2x_3 + y_3x_1) \right| \]Plug in the points:\[ \text{Area} = \frac{1}{2} \left| 0(-1) + 4(3) + 2(1) - (1 \times 4 + (-1) \times 2 + 3 \times 0) \right| \]Simplify:\[= \frac{1}{2} \left| 0 + 12 + 2 - (4 - 2 + 0) \right| \]\[= \frac{1}{2} \left| 14 - 2 \right| \]\[= \frac{1}{2} \times 12 = 6 \]

Step 4: Conclusion: State the final area

Therefore, the area of the region bounded by the three lines is 6 square units.

Key Concepts

Intersection Points

Intersection points are where two or more curves or lines meet. These points help us define the boundary of a region. To find the intersection points for lines or curves, you solve their equations simultaneously.
When dealing with the problem of finding the area of a region bounded by graphs, identifying these intersection points is a crucial initial step. It's here we establish the vertices of the polygon (often a triangle) whose area is needed. For the given problem, we found the intersections by equating pairs of line equations and solving for the variables.

• First, for the lines \(y = -\frac{1}{2}x + 1\) and \(y = x + 1\), we found the intersection point \((0, 1)\) by setting the equations equal and solving for \(x\) and \(y\).
• Second, intersecting \(y = -\frac{1}{2}x + 1\) and \(y = -2x + 7\), we got the point \((4, -1)\).
• Finally, for \(y = x + 1\) and \(y = -2x + 7\), the intersection point is \((2, 3)\).

These intersection points will later be used to calculate the area within the bounded region using a specific formula.

Shoelace Formula

The shoelace formula, also known as Gauss's area formula, is a mathematical algorithm to determine the area of a polygon when its vertices' coordinates are known. For students tackling the problem given, after finding the intersection points \((0, 1), (4, -1), (2, 3)\), this formula comes in handy.
The shoelace formula is defined as: \[ \text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_1 - (y_1x_2 + y_2x_3 + y_3x_1) \right|
\]Here's how it works:

• Multiply the \(x\) coordinate of each vertex by the \(y\) coordinate of the next vertex (moving clockwise), sum these products.
• Then, multiply the \(y\) coordinate of each vertex by the \(x\) coordinate of the next vertex (again moving clockwise), and sum these values.
• Subtract the second sum from the first, take the absolute value, and divide by 2 to get the polygon's area.

For the triangle formed by points \((0, 1), (4, -1), (2, 3)\), using this formula gives the area as 6 square units. This process effectively transforms the problem of finding an area into straightforward algebraic steps.

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a straight line. It is one of the most common and useful formats in algebra. In this form, the equation is expressed as \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) is the y-intercept.
The slope of a line indicates its steepness and direction, while the y-intercept is where the line crosses the y-axis. For solving our exercise, converting given equations to this form made it easy to see how the lines relate to each other.

• For \(x + 2y = 2\), rearranging gives \(y = -\frac{1}{2}x + 1\).
• Converting \(y - x = 1\), results in \(y = x + 1\).
• For \(2x + y = 7\), we find \(y = -2x + 7\).

Each conversion provides clear insight into the line's behavior just by looking at its equation. This aids not only in visualization but also in computation, like identifying intersection points, which promote further solutions such as area calculations.