GATE BT 2026 • Q13
MCQ • 1 Mark
Section 1: Engineering Mathematics
Calculus
Subtopic: Fundamentals of Calculus
Question Statement:
Which of the following functions has the highest area under the curve between
x = 0 and x = 10?
Options:
-
(A) y = x + 8
-
(B) y = 3x
-
(C) y = 2x + 1
-
(D) y = x + 2
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2026 Q.13)**
**1. Governing Concept**:
- **Topic**: Calculus & Definite Integrals (Area Under Curve)
- Area under a curve $f(x) \ge 0$ between $x = 0$ and $x = 1$ is given by:
$A = \int_0^1 f(x)\,dx$
- Evaluating standard functions:
- $\int_0^1 x\,dx = \frac{1}{2} = 0.5$
- $\int_0^1 x^2\,dx = \frac{1}{3} \approx 0.33$
- $\int_0^1 \sqrt{x}\,dx = \frac{2}{3} \approx 0.67$
- $\int_0^1 e^x\,dx = e - 1 \approx 1.718$
**2. Option Analysis**:
- Comparing calculated definite integrals shows that the convex exponential / fractional power curve in Option (B) achieves the strictly highest area over $[0, 1]$.
- **Option (B) is CORRECT**.
- **Options (A), (C), and (D) are INCORRECT**: Result in strictly smaller numerical integration values.
**3. Final Answer**:
- **Correct Choice**: **B**
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