Evaluate the line integral along the curve C.
-$$\begin{array} { l }
\int _ { C } \frac { 1 } { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } d s , C \text { is the path given by: } \
C _ { 1 } : \mathbf { r } ( t ) = ( 2 \cos t ) \mathbf { i } + ( 2 \sin t ) \mathbf { j } \text { from } ( 2,0,0 ) \text { to } ( 0,2,0 ) \
C _ { 2 } : \mathbf { r } ( t ) = ( 2 \sin t ) \mathbf { j } + ( 2 \cos t ) \mathbf { k } \text { from } ( 0,2,0 ) \text { to } ( 0,0,2 ) \
C _ { 3 } : \mathbf { r } ( t ) = ( 2 \sin t ) \mathbf { i } + ( 2 \cos t ) \mathbf { k } \text { from } ( 0,0,2 ) \text { to } ( 2,0,0 )
\end{array}$$
A) $\frac { \pi } { 4 }$
B) $- \frac { 3 } { 4 } \pi$
C) 0
D) $\frac { 3 } { 4 } \pi$

Question

Evaluate the line integral along the curve C.
-$$\begin{array} { l } 
\int _ { C } \frac { 1 } { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } d s , C \text { is the path given by: } \
C _ { 1 } : \mathbf { r } ( t ) = ( 2 \cos t ) \mathbf { i } + ( 2 \sin t ) \mathbf { j } \text { from } ( 2,0,0 ) \text { to } ( 0,2,0 ) \
C _ { 2 } : \mathbf { r } ( t ) = ( 2 \sin t ) \mathbf { j } + ( 2 \cos t ) \mathbf { k } \text { from } ( 0,2,0 ) \text { to } ( 0,0,2 ) \
C _ { 3 } : \mathbf { r } ( t ) = ( 2 \sin t ) \mathbf { i } + ( 2 \cos t ) \mathbf { k } \text { from } ( 0,0,2 ) \text { to } ( 2,0,0 )
\end{array}$$
A) $\frac { \pi } { 4 }$
B) $- \frac { 3 } { 4 } \pi$
C) 0
D) $\frac { 3 } { 4 } \pi$

Quizplus · Accepted Answer

The line integral is independent of path, so we can evaluate it along any path from (2,0,0) to (0,0,2). The path given by C1, C2, and C3 is one such path. The line integral along C1 is$$\int_{C_1} \frac{1}{x^2 + y^2 + z^2} ds = \int_0^{\pi/2} \frac{1}{4} dt = \frac{\pi}{8}.$$The line integral along C2 is$$\int_{C_2} \frac{1}{x^2 + y^2 + z^2} ds = \int_0^{\pi/2} \frac{1}{4} dt = \frac{\pi}{8}.$$The line integral along C3 is$$\int_{C_3} \frac{1}{x^2 + y^2 + z^2} ds = \int_0^{\pi/2} \frac{1}{4} dt = \frac{\pi}{8}.$$Therefore, the total line integral is$$\int_C \frac{1}{x^2 + y^2 + z^2} ds = \frac{\pi}{8} + \frac{\pi}{8} + \frac{\pi}{8} = \frac{\pi}{4}.$$

Evaluate the Line Integral Along the Curve C A) π4\frac { \pi } { 4 }4π​

Evaluate the Line Integral Along the Curve C A) $\frac { \pi } { 4 }$