Question

Find the area between the line $ y = x - 1 $, $ y = 0 $, $ -2 \leq y \leq 2 $.

• A. 6
• B. 8
• C. 4
• D. 10

Hint:

To find the area between a line and the x-axis, set up an integral with the line equation and integrate over the given limits.

Answer 1

The Correct Option is C

We are asked to find the area between the line \( y = x - 1 \), \( y = 0 \), and the interval \( -2 \leq y \leq 2 \). 
First, express \( y = x - 1 \) in terms of \( x \): \[ x = y + 1 \] Now, the limits of integration are \( y = -2 \) and \( y = 2 \). 
The area is found by integrating the expression for \( x \): \[ \text{Area} = \int_{-2}^{2} (y + 1) \, dy \] Evaluating the integral: \[ \text{Area} = \left[ \frac{y^2}{2} + y \right]_{-2}^{2} \] Substitute the limits: \[ = \left( \frac{2^2}{2} + 2 \right) - \left( \frac{(-2)^2}{2} + (-2) \right) \] \[ = \left( \frac{4}{2} + 2 \right) - \left( \frac{4}{2} - 2 \right) \] \[ = (2 + 2) - (2 - 2) \] \[ = 4 \] 
Thus, the area is \( 4 \).