Word Count : 278

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We are tasked with evaluating the integral: \[ \int_C (2x^2 + 3y^2) \, dx \] where \( C \) is the curve parameterized by: \[ x(t) = at^2, \quad y(t) = 2at, \quad 0 \leq t \leq 1 \] ### Step-by-Step Solution: 1. Find the expression for \( dx \): \[ x(t) = at^2 \quad \Rightarrow \quad \frac{dx}{dt} = 2at \] So: \[ dx = 2at \, dt \] ________ _________ __________ ______ ________ _________ ______ __________ ____ ______.
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