Define [x] as the greatest integer less than or equal to x, for each x ∈ (-∞, ∞). If y = [x], then area under y for x ∈ [1, 4] is
(A) 1
(B) 3
(C) 4
(D) 6
Options:
(A) 1
(B) 3
(C) 4
(D) 6
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2020 Q.5 / GA Q.5)**
**1. Mathematical Formulation**:
- The greatest integer function $y = [x]$ is a piecewise constant step function.
- The area under the curve from $x = 1$ to $x = 4$ is given by the definite integral:
$\text{Area} = \int_{1}^{4} [x] \, dx$
**2. Step-by-Step Piecewise Integration**:
- Splitting the integral across consecutive unit intervals:
$\text{Area} = \int_{1}^{2} [x] \, dx + \int_{2}^{3} [x] \, dx + \int_{3}^{4} [x] \, dx$
- For $x \in [1, 2)$, $[x] = 1 \implies \int_{1}^{2} 1 \, dx = 1 \times (2 - 1) = 1$
- For $x \in [2, 3)$, $[x] = 2 \implies \int_{2}^{3} 2 \, dx = 2 \times (3 - 2) = 2$
- For $x \in [3, 4)$, $[x] = 3 \implies \int_{3}^{4} 3 \, dx = 3 \times (4 - 3) = 3$
- Summing the areas:
$\text{Area} = 1 + 2 + 3 = 6$
**3. Official Verification**:
The verified official IIT Delhi Master Answer is **Option (D)** (`6`).
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