Answer :
Final answer:
To integrate f(x,y) =x+y over the curve c: x² + y² =49 requires substitution method in parameters as well as integral calculus. The integral simplifies to 49[π/2 - 1].
Explanation:
To solve the problem, we need to parameterize the curve C given by the equation x² + y² =49 in the first quadrant and then compute the line integral of the scalar field f(x,y) = x + y. Let's parameterize the curve as x=7cos(t), y=7sin(t) where the parameter t goes from 0 to π/2.
Now, using these values, the differential element ds is sqrt((dx/dt)²+(dy/dt)²)dt = 7dt. The function f(x,y) now becomes f(7cos(t), 7sin(t)) = 7cos(t) + 7sin(t). So, the line integral ∫(x+y)ds over the curve c from (7,0) to (0,7) becomes ∫ from 0 to π/2 [7cos(t) + 7sin(t)]*7 dt = 49∫ from 0 to π/2 [cos(t) + sin(t)] dt.
Integrating this, we find that the integral simplifies to 49[ t - cos(t)] from 0 to π/2, which gives us the solution 49[π/2 - 1].
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