Question

The area of the region bounded by the lines, $y = x + 2$, $x = 0$, $x = 1$ and $y = 0$ is

• A. $2$ sq.units
• B. $\frac{5}{2}$ sq.units
• C. $\frac{9}{2}$ sq.units
• D. $9$ sq.units
• E. $12$ sq.units

Hint:

For area problems: - Always identify upper and lower curves carefully - Limits usually come from vertical boundaries ($x = a$ to $x = b$)

Answer 1

The Correct Option is B

Concept: Area between curves is given by: \[ \text{Area} = \int_{a}^{b} (\text{upper function} - \text{lower function})\,dx \]

Step 1: Identify the region and limits.
The region is bounded by: \[ x = 0,\; x = 1 \] Upper curve: $y = x + 2$
Lower curve: $y = 0$

Step 2: Set up the integral.
\[ \text{Area} = \int_{0}^{1} \left[(x + 2) - 0\right] dx \]

Step 3: Evaluate the integral.
\[ = \int_{0}^{1} (x + 2)\,dx = \left[\frac{x^2}{2} + 2x\right]_0^1 \] \[ = \left(\frac{1}{2} + 2\right) - 0 = \frac{5}{2} \]