Question

The value of the integral $\int_0^4 (x - f(x))\,dx$, where 
 

is:
 

• A. 2
• B. 1
• C. –1
• D. –2

Hint:

For piecewise constant functions, integrate each segment separately. The total integral is simply the sum of areas under each linear section.

Answer 1

The Correct Option is C

Step 1: Break the integral into parts.
\[ \int_0^4 (x - f(x)) dx = \int_0^1 (x - 0) dx + \int_1^2 (x - 1) dx + \int_2^3 (x - 2) dx + \int_3^4 (x - 3) dx \] Step 2: Compute each term.
\[ \int_0^1 x\,dx = \frac{1}{2} \] \[ \int_1^2 (x - 1) dx = \frac{1}{2} \] \[ \int_2^3 (x - 2) dx = \frac{1}{2} \] \[ \int_3^4 (x - 3) dx = \frac{1}{2} \] Step 3: Add results.
\[ I = \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} = 2 \] Step 4: Conclusion.
Therefore, the value of the integral is 2.