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[(xy)ds_C: r(t) = (2cos(t), 2sin(t)), 0≤t≤π/2 C

[(xy)ds_C: r(t) = (2cos(t), 2sin(t)), 0≤t≤π/2 C

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem 2:** Evaluate the line integral \(\int_{C} (xy) \, ds\), where the curve \( C \) is parameterized by \(\mathbf{r}(t) = \langle 2\cos(t), 2\sin(t) \rangle\) for \(0 \leq t \leq \pi/2\). ### Explanation of Graph or Diagram This problem involves calculating a line integral along a curve \( C \) in the plane. The parameterization \(\mathbf{r}(t) = \langle 2\cos(t), 2\sin(t) \rangle\) describes a segment of a circle with radius 2 centered at the origin. The curve \( C \) starts at the point \((2, 0)\) when \( t = 0 \) and ends at \((0, 2)\) when \( t = \pi/2 \). The interval \(0 \leq t \leq \pi/2\) indicates that this is the first quadrant of the circle. This mathematical representation is used to evaluate the integral along the specified path.
Transcribed Image Text:**Problem 2:** Evaluate the line integral \(\int_{C} (xy) \, ds\), where the curve \( C \) is parameterized by \(\mathbf{r}(t) = \langle 2\cos(t), 2\sin(t) \rangle\) for \(0 \leq t \leq \pi/2\). ### Explanation of Graph or Diagram This problem involves calculating a line integral along a curve \( C \) in the plane. The parameterization \(\mathbf{r}(t) = \langle 2\cos(t), 2\sin(t) \rangle\) describes a segment of a circle with radius 2 centered at the origin. The curve \( C \) starts at the point \((2, 0)\) when \( t = 0 \) and ends at \((0, 2)\) when \( t = \pi/2 \). The interval \(0 \leq t \leq \pi/2\) indicates that this is the first quadrant of the circle. This mathematical representation is used to evaluate the integral along the specified path.
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