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

The Correct Option is B

The problem requires finding the area bounded by the curve \( y = |x - 2| \), the x-axis (y = 0), x = 0, and x = 5. 

First, analyze the curve:

\(y = |x - 2|\)
1. When \(x < 2\), \(y = 2 - x\)
2. When \(x \geq 2\), \(y = x - 2\)

The critical point is at \(x = 2\). The expression \(y = |x - 2|\) forms a V-shaped graph.

Calculate the area between x = 0 and x = 5 by dividing it into two regions using x = 2.

For \(0 \leq x < 2\):
Area under \(y = 2 - x\)
\[ \int_{0}^{2} (2 - x) \, dx = \left[ 2x - \frac{x^2}{2} \right]_{0}^{2} = \left(4 - 2\right) = 2 \]
For \(2 \leq x \leq 5\):
Area under \(y = x - 2\)
\[ \int_{2}^{5} (x - 2) \, dx = \left[ \frac{x^2}{2} - 2x \right]_{2}^{5} = \left(\frac{25}{2} - 10\right) - \left(\frac{4}{2} - 4\right) = \frac{9}{2} \]

Summing both areas gives:

Final Area = \( 2 + \frac{9}{2} = \frac{4}{2} + \frac{9}{2} = \frac{13}{2} = 6.5 \) square units.

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

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.