Problem 40 - College Algebra and Calculus: An Applied Approach

Question

Use integration to find the area of the triangular region having the given vertices. $$ (0,0),(4,0),(6,4) $$

Step-by-Step Solution

Answer

The area of the triangle is 16 unit square.

Step 1: Identify the vertices

Identify the vertices of the triangle. Here the vertices are given as (0,0), (4,0) and (6,4). Two of the vertices lie on x-axis and the third vertex tell us about the height of the triangle.

Step 2: Calculate the slope

Calculate the slope of the line passing through the points (4,0) and (6,4). The formula for slope (\(m\)) is given by \(m = \frac{y2 - y1}{x2 - x1}\). Substituting the coordinates of the two points, we get \(m = \frac{4 - 0}{6 - 4} = 2\)

Step 3: Find the equation of the line

Form equation of line passing through the points (4,0) and (6,4) which is given by \(y = mx + c\), substituting the values we get \(y = 2x -8\).

Step 4: Calculate the area using integration

The area of triangle can be found by integrating \(y = 2x -8\) from 0 to 4. Thus, the area (\(A\)) of triangle is given by \(A= \int_{0}^{4} (2x -8) dx\). On integrating, we get \(A = [x^2 - 8x]_{0}^{4} = (4^2 - 4*8) - (0) = 16 - 32 = -16 \)

Step 5: Calculate Absolute Value of Area

Area is always a positive value but the integration gives negative value because the triangle lies below the x-axis. Thus, we take the absolute value of this area. So, \( |A| = |-16| = 16 \)