Question

Area bounded by y=|x-5| and x-axis between x = 2; and x = 4 is:

• A. 5
• B. 4
• C. 3
• D. 6

Answer 1

The Correct Option is B

Area bounded by \(y=|x-5|\) and x-axis between \(x = 2\) and \(x = 4\) is:

To find the area bounded by the curve \(y=|x-5|\) and the x-axis between \(x=2\) and \(x=4\), we first need to understand the behavior of the absolute value function \(y=|x-5|\).

This function can be rewritten in piecewise form as:
\(y=x-5\) when \(x\geq5\) and \(y=-(x-5)=5-x\) when \(x<5\).

Since \(2\leq x\leq4\), in this interval \(x<5\), so we use \(y=5-x\).

We can now find the area under the curve \(y=5-x\) from \(x=2\) to \(x=4\) by calculating the definite integral:

\(\text{Area} = \int_{2}^{4}(5-x) \, dx\)

Calculate the integral: \(\int(5-x) \, dx = 5x - \frac{x^2}{2} + C\).

Evaluate it from 2 to 4:

\(\text{Area} = \left[5x - \frac{x^2}{2}\right]_{2}^{4}\)

\(\text{Area} = (5(4) - \frac{4^2}{2}) - (5(2) - \frac{2^2}{2})\)

\(\text{Area} = (20 - 8) - (10 - 2)\)

\(\text{Area} = 12 - 8\)

\(\text{Area} = 4\)

The area bounded by \(y=|x-5|\) and the x-axis from \(x=2\) to \(x=4\) is \(\boxed{4}\).