Question

The area (in sq. units) of the region bounded by the curve \( y = x \), x-axis, \( x = 0 \) and \( x = 2 \) is:

• (A) \( rac{3}{2} \)
• (B) \( rac{\log 2}{2} \)
• (C) 2
• (D) 4

Answer 1

The Correct Option is C

Step 1: Define the problem
Determine the area enclosed by the function y = x, the x-axis, and the vertical lines x = 0 and x = 2.

Step 2: Formulate the integral
The area is represented by the definite integral of the function y = x from x = 0 to x = 2:
Area = ∫₀² x dx

Step 3: Integrate the function
The antiderivative of x is (x²)/2.
Area = [ (x²)/2 ] evaluated from 0 to 2.

Step 4: Evaluate the definite integral
Apply the limits of integration:
Upper limit (x = 2): (2)² / 2 = 4 / 2 = 2
Lower limit (x = 0): (0)² / 2 = 0
Area = 2 - 0 = 2 square units.

Step 5: State the result
The area of the specified region is 2 square units.

Final Answer: (C) 2