Question

(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

Answer 1

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
`\( f \)`.