Question

Use integration to find the area of the figure having the given vertices.
(0, 0), (1, 6), (3, −2), (1, -7)

Answer 1

The figure with vertices $(0, 0)$, $(1,6)$, $(3, -2)$ and $(1,-7)$ has an area of 16.5 square units.

How to obtain the area of a figure by integration

With the support of a graphic tool, we have the following figure shown in the image attached below. According to definite integration, there are two characteristics on area calculation to be taken into account:

1. The area above the curve is positive if the curve is above x-axis.
2. The area below the curve is negative if the curve is above y-axis.
3. If the curve is a straight line, then the integral of the curve is determined by triangle formula.

Then, we determine the area by using the following calculation:

$A = \frac{1}{2}\cdot (2.5-1)\cdot (6-0) + \frac{1}{2}\cdot (1-0)\cdot (6-0)-\frac{1}{2}\cdot (1-0)\cdot (-7-0) - \frac{1}{2}\cdot (3-1) \cdot [-7-(-2)]-\frac{1}{2}\cdot (3-2.5)\cdot (-2-0)$

$A = 16.5$

The figure with vertices $(0, 0)$, $(1,6)$, $(3, -2)$ and $(1,-7)$ has an area of 16.5 square units. $\blacksquare$

To learn more on definite integrals, we kindly invite to check this verified question: https://brainly.com/question/22655212