Answer :
The work done by the force field F along the curve C, given the force field, is C. 145.
How to find the work done ?
In this case, the force field is F(x, y) = ⟨5x, 5y⟩ and the curve C is the line segment from (2, 1) to (3, 5).
The vector element along the curve can be written as:
dr = (dx, dy)
Substituting the force field and vector element into the line integral, we get:
∫CF · dr = ∫(5x, 5y) · (dx, dy)
We can now evaluate the line integral. We can parameterize the line segment C by x = 2 + t, y = 1 + 4t, where 0 ≤ t ≤ 1.
Substituting these into the line integral, we get:
∫CF · dr = ∫0¹ (5(2 + t), 5(1 + 4t)) · (dx, dy)
= ∫0¹ (10 + 5t, 5 + 20t) · (dx, dy)
= ∫0¹ 10 + 5t + 5 + 20t dt
= ∫0¹ 35t + 15 dt
= 145
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