Problem 286 - Calculus Volume 1

Question

For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area \(M\) and the centroid \((\overline{x}, \overline{y})\) for the given shapes. Use symmetry to help locate the center of mass whenever possible. Triangle: \(y=x, \quad y=2-x,\) and \(y=0\)

Step-by-Step Solution

Answer

Area \(M = 1\), Centroid \((\overline{x}, \overline{y}) = (1, \frac{1}{3})\).

Step 1: Identify the region

The given lines are \(y = x\), \(y = 2-x\), and \(y = 0\). These lines form a triangle in the xy-plane. The points of intersection can be found as \((0,0)\), \((1,1)\), and \((2,0)\). Thus, the triangle is bounded by these vertices.

Step 2: Draw the region

Using a graphing calculator or plotting software, graph the lines \(y = x\), \(y = 2-x\), and \(y = 0\). Shade the triangular region bounded by these lines. This region is a right triangle with vertices at \((0,0)\), \((1,1)\), and \((2,0)\).

Step 3: Calculate the area of the triangle

The area \(M\) of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) can be calculated using the formula:\[ M = \frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)| \]Substituting the vertices \((0,0)\), \((1,1)\), and \((2,0)\), we get:\[ M = \frac{1}{2} |0(1-0) + 1(0-0) + 2(0-1)| = \frac{1}{2} |0 + 0 - 2| = \frac{1}{2} \times 2 = 1 \]

Step 4: Find the centroid \(\overline{x}\)

The centroid \((\overline{x}, \overline{y})\) of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given by:\[ \overline{x} = \frac{x_1 + x_2 + x_3}{3}, \overline{y} = \frac{y_1 + y_2 + y_3}{3} \]Substituting the vertices \((0,0)\), \((1,1)\), and \((2,0)\), we get:\[ \overline{x} = \frac{0 + 1 + 2}{3} = \frac{3}{3} = 1 \]

Step 5: Find the centroid \(\overline{y}\)

Using the same formula for the y-coordinate:\[ \overline{y} = \frac{0 + 1 + 0}{3} = \frac{1}{3} \]

Step 6: Use symmetry to verify centroid

The triangle is symmetric about the line \(x=1\). The calculated centroid \((1, \frac{1}{3})\) lies on this line, confirming that the calculations were done correctly, and the center of mass is at the centroid due to symmetry.

Key Concepts

Area of a Triangle

The area of a triangle can be easily found if you know its vertices. This is particularly handy in coordinate geometry. To find the area from the coordinates, there's a straightforward formula. It's:\[ M = \frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)| \]Here, \(x_1, y_1\), \(x_2, y_2\), and \(x_3, y_3\) are the coordinates of the vertices.

• First, plug in your known vertex values.
• Perform the calculations inside the absolute value function first.
• Then divide the result by 2.
• This gives you the area, and remember: areas are always positive!

For example, in the given exercise with vertices \(0, 0\), \(1, 1\), and \(2, 0\), the calculation shows this triangle area as 1 square unit.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows the representation of geometric figures in a numerical context. This approach is powerful for solving problems involving shapes on a coordinate plane. You can analyze various properties using coordinate geometry:

• Find intersections of lines by solving equations.
• Calculate distances between points using the distance formula.
• Determine slopes of lines for parallelism or perpendicularity.
• Utilize formulas to find midpoints or centroids, like with triangles.

In our exercise example, using coordinate geometry, we find the centroid of a triangle by averaging its vertices coordinates:\[ \overline{x} = \frac{x_1 + x_2 + x_3}{3} \] and \[ \overline{y} = \frac{y_1 + y_2 + y_3}{3} \]. This straightforward calculation gives the center point, or centroid, accurately.

Symmetry in Geometry

Symmetry in geometry means a shape looks the same even after certain transformations like reflection or rotation. Symmetry can offer shortcuts in solving geometry problems by reducing complex situations into simpler ones. Consider the triangle in the example:

• This triangle is symmetrical about the line \(x=1\), meaning it's mirrored across this line.
• The symmetry helps ensure that the centroid appears correctly balanced around the midpoint of this line.
• Due to symmetry, you can often use the geometrical balance to guess or verify results like centroids before computation.

Understanding symmetry will allow you to informally check your results or simplify complex calculations, confirming that centroids, areas, and other geometric properties fit the expected patterns each time.