Question

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

Short answer

The area of the triangle is 16 unit square.

Step-by-step solution

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.

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 \(m = \frac{4 - 0}{6 - 4} = 2\)

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 \(y = 2x -8\).

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 \(A = [x^2 - 8x]_{0}^{4} = (4^2 - 4*8) - (0) = 16 - 32 = -16 \)

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, \( |A| = |-16| = 16 \)