GATE 2024 CH – Question 51
Consider the line integral $\int_C\mathbf{F}(\mathbf{r}) \cdot d\mathbf{r}$, with $\mathbf{F}(\mathbf{r}) = x\hat{i} + y\hat{j} + z\hat{k}$, where $\hat{i}$, $\hat{j}$ and $\hat{k}$ are unit vectors in the $(x, y, z)$ Cartesian coordinate system. The path C is given by $\mathbf{r}(t) = \cos(t)\hat{i} + \sin(t)\hat{j} + t\hat{k}$, where $0 \le t \le \pi$. The value of the integral, rounded off to 2 decimal places, is _________
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Correct answer: 4.91 to 4.95
Explanation
The field is the gradient of $\frac{1}{2}(x^2 + y^2 + z^2)$, so the integral depends only on the end points: from $t = 0$, the point $(1, 0, 0)$, to $t = \pi$, the point $(-1, 0, \pi)$. The value is $\frac{1}{2}\left[(1 + \pi^2) - 1\right] = \frac{\pi^2}{2} = 4.93$.