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: Understand the problem
We need to find the area of the region bounded by the curve y = x, the x-axis, and the vertical lines x = 0 and x = 2.

Step 2: Set up the integral for the area
The area under the curve y = x from x = 0 to x = 2 is given by the definite integral:
Area = ∫ from 0 to 2 of x dx

Step 3: Calculate the integral
The integral of x with respect to x is (x²)/2.
So, Area = [ (x²)/2 ] evaluated from 0 to 2.

Step 4: Evaluate the definite integral
Substitute the upper limit x = 2:
(2)² / 2 = 4 / 2 = 2
Substitute the lower limit x = 0:
(0)² / 2 = 0
So, Area = 2 - 0 = 2 square units.

Step 5: Conclusion
The area of the region bounded by the curve y = x, the x-axis, x = 0 and x = 2 is 2 square units.

Final Answer: (C) 2