Evaluate the line integral over the given curve C. $\int _ { C } \left( 3 x + 7 y ^ { 3 } \right) d s$ ; $C : \mathbf { r } ( t ) = ( t - 3 ) \mathbf { i } + t \mathbf { j }$ , $0 \leq t \leq 3$
A) $27$
B) $\frac { 513 } { 4 }$ $\sqrt { 2 }$
C) $36$
D) $72$ $\sqrt { 2 }$

Question

Quizplus · Accepted Answer

The line integral $\int _ { C } \left( 3 x + 7 y ^ { 3 } \right) d s$ is evaluated using the parameterization $\mathbf { r } ( t ) = ( t - 3 ) \mathbf { i } + t \mathbf { j }$. The differential arc length $ds$ is $\sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt = \sqrt{2} dt$. Substituting $x = t - 3$ and $y = t$ into the integrand and integrating from $t = 0$ to $t = 3$ gives the result $\frac { 513 } { 4 }$ $\sqrt { 2 }$.

Evaluate the Line Integral Over the Given Curve C ∫C(3x+7y3)ds\int _ { C } \left( 3 x + 7 y ^ { 3 } \right) d s∫C​(3x+7y3)ds

Evaluate the Line Integral Over the Given Curve C $\int _ { C } \left( 3 x + 7 y ^ { 3 } \right) d s$