Evaluate the line integral of the function over the given curve: $f ( x , y ) = x$ and the curve C is given by $y = x ^ { 2 } , 0 \leq x \leq 2$
A) $\frac { 17 \sqrt { 17 } - 1 } { 12 }$
B) $\frac { 17 \sqrt { 17 } - 1 } { 14 }$
C) $\frac { 17 \sqrt { 17 } - 1 } { 17 }$
D) $\frac { 17 \sqrt { 17 } - 2 } { 12 }$

Question

Quizplus · Accepted Answer

To evaluate the line integral of $f(x, y) = x$ over the curve $y = x^2$ from $x = 0$ to $x = 2$, we parameterize the curve using $x$ itself. Thus, $y = x^2$ and the differential arc length $ds$ can be expressed in terms of $x$ as $ds = \sqrt{1 + (dy/dx)^2}dx = \sqrt{1 + (2x)^2}dx$. The line integral becomes $\int_C f(x, y) ds = \int_0^2 x \sqrt{1 + 4x^2} dx$. Solving this integral gives $\frac{17\sqrt{17} - 1}{12}$.

Evaluate the Line Integral of the Function Over the Given f(x,y)=xf ( x , y ) = xf(x,y)=x

Evaluate the Line Integral of the Function Over the Given $f ( x , y ) = x$