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(1 point) Show F(x, y) = (6xy+ 5)i + (9x²y2 + 2e2y); is conservative by finding a potential function f for F, and use f to compute F. dr, where C is the curve given by lo r(t) = (2 sin’t)i + (*sin"(t)); 2t ) + — 3t for 0

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Final Answer:

The potential function for
\( F(x, y) = (6xy + 5)i + (9x^2y^2 + 2e^(2y)) \) is \( f(x, y) = 3x^2y + 5x + e^(2y) + C \), where \( C \) is a constant. Using this potential function, the line integral of
\( F \) along the curve
\( C \) is \( f(r(b)) - f(r(a)) = 4 \sin^2(b) - 4 \sin^2(a) + 3b^2 - 3a^2 + 5(b - a) + e^(2b) - e^(2a) \).

Step-by-step explanation:

To determine if
\( F \) is conservative, we find a potential function
\( f \) such that \( \\abla f = F \), where \( \\abla \) denotes the gradient. In this case,
\( F(x, y) = (6xy + 5)i + (9x^2y^2 + 2e^(2y)) \). Integrating each component with respect to
\( x \) and \( y \) separately, we find \( f(x, y) = 3x^2y + 5x + e^(2y) + C \), where
\( C \) is a constant.

To compute
\( \int_C F \cdot dr \), we evaluate \( f \) at the endpoints of \( C \), given by
\( r(a) \) and \( r(b) \), and then subtract the results. The curve c is parameterized by
\( r(t) = (2\sin t)i + (t^2 - 3t)j \), where \( 0 \leq t \leq \pi \).The final expression
\( 4 \sin^2(b) - 4 \sin^2(a) + 3b^2 - 3a^2 + 5(b - a) + e^(2b) - e^(2a) \) is the result of plugging in the values from
\( r(a) \) to \( r(b) \). This confirms the conservative nature of F, as the line integral depends only on the endpoints, validating the existence of the potential function
\( f \).

User Andrew Quebe
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