Evaluate the line integral of f(x,y) along the curve C.
-$$f ( x , y ) = x ^ { 2 } + y ^ { 2 } , C : y = - 4 x - 3,0 \leq x \leq 3$$
A) $96 \sqrt { 17 }$
B) $594 \sqrt { 17 }$
C) $288 \sqrt { 17 }$
D) 288

Question

Quizplus · Accepted Answer

We first parameterize the curve C using x as the parameter:
$$x = x$$
$$y = -4x-3$$
$$0 \leq x \leq 3$$

Then, we can calculate the line integral as:
$$\int_C f(x,y) ds = \int_0^3 [(x^2+(-4x-3)^2)\sqrt{1+(-4)^2}] dx$$
$$\int_0^3 [(x^2+16x^2+24x+9)\sqrt{17}] dx$$
$$\int_0^3 [(17x^2+24x+9)\sqrt{17}] dx$$
$$[\frac{17}{3}x^3+12x^2+9x\sqrt{17}]_0^3$$
$$288\sqrt{17}$$

Therefore, the answer is C, 288$\sqrt{17}$.

Evaluate the Line Integral of F(x,y) Along the Curve C f(x,y)=x2+y2,C:y=−4x−3,0≤x≤3f ( x , y ) = x ^ { 2 } + y ^ { 2 } , C : y = - 4 x - 3,0 \leq x \leq 3f(x,y)=x2+y2,C:y=−4x−3,0≤x≤3

Evaluate the Line Integral of F(x,y) Along the Curve C $f ( x , y ) = x ^ { 2 } + y ^ { 2 } , C : y = - 4 x - 3,0 \leq x \leq 3$