Question

The area (in square units) bounded by the curve y = |x−2| between x = 0, y = 0, and x = 5 is:

• A. 8
• B. 6.5
• C. 13
• D. 3.5

Answer 1

Answer: (B)

6.5

Answer 2

The curve $y=|x-2|$ is split into two linear parts:

• For $x \geq 2: y = x - 2$
• For $x < 2: y = 2 - x$

We calculate the area under the curve from $x = 0$ to $x = 5$, divided into two regions:

1. From $x = 0$ to $x = 2: (y = 2 - x)$
2. From $x = 2$ to $x = 5: (y = x - 2)$

Region 1: $x \in [0, 2]$ The area under $y = 2 - x$ is:

$Area_1 = \int_0^2 (2 - x) dx.$

$Area_1 = \left[ 2x - \frac{x^2}{2} \right]_0^2 = \left( 2(2) - \frac{2^2}{2} \right) - \left( 2(0) - \frac{0^2}{2} \right).$

$Area_1 = (4 - 2) - 0 = 2.$

Region 2: $x \in [2, 5]$ The area under $y = x - 2$ is:

$Area_2 = \int_2^5 (x - 2) dx.$

$Area_2 = \left[ \frac{x^2}{2} - 2x \right]_2^5 = \left( \frac{5^2}{2} - 2(5) \right) - \left( \frac{2^2}{2} - 2(2) \right).$

$Area_2 = \left( \frac{25}{2} - 10 \right) - \left( \frac{4}{2} - 4 \right).$

$Area_2 = \left( \frac{25}{2} - \frac{20}{2} \right) - \left( \frac{4}{2} - \frac{8}{2} \right) = \frac{5}{2} + \frac{4}{2} = \frac{9}{2}.$

Total Area:

$Total Area = Area_1 + Area_2 = 2 + \frac{9}{2} = \frac{4}{2} + \frac{9}{2} = \frac{13}{2} = 6.5.$

Thus, the area bounded by the curve is 6.5 square units.