Question

Two straight lines pass through the origin \( (x_0, y_0) = (0, 0) \). One of them passes through the point \( (x_1, y_1) = (1, 3) \) and the other passes through the point \( (x_2, y_2) = (1, 2) \). What is the area enclosed between the straight lines in the interval \( [0, 1] \) on the x-axis?

• A. 0.5
• B. 1.0
• C. 1.5
• D. 2.0

Hint:

To calculate the area between two curves, subtract one curve's equation from the other and integrate over the given interval.

Answer 1

The Correct Option is A

To solve this problem, we need to calculate the area between the two lines in the interval \( [0, 1] \) on the x-axis. Step 1: Equation of the lines.
- Line 1 (through (0, 0) and (1, 3)): The slope of the line is: \[ m_1 = \frac{3 - 0}{1 - 0} = 3. \] The equation of the line is: \[ y_1 = 3x. \] - Line 2 (through (0, 0) and (1, 2)): The slope of the line is: \[ m_2 = \frac{2 - 0}{1 - 0} = 2. \] The equation of the line is: \[ y_2 = 2x. \] Step 2: Calculate the area between the lines.
The area between the lines is given by the integral of the difference in the y-values of the two lines over the interval \( [0, 1] \): \[ \text{Area} = \int_0^1 (y_1 - y_2) \, dx = \int_0^1 (3x - 2x) \, dx = \int_0^1 x \, dx. \] The integral is: \[ \int_0^1 x \, dx = \frac{x^2}{2} \Big|_0^1 = \frac{1}{2}. \] Thus, the area is \( 0.5 \). Therefore, the correct answer is (A).