Question

Evaluate the line integral, where C is the given curve. z2 dx + x2 dy + y2 dz, C C is the line segment from (1, 0, 0) to (4, 1, 3)

Answer 1

The line integral is the path of the function along a line having a continuous value.

The integral solved gives the value 111.

The given question is

$\int\limits^c {z}^2 \, dx+ {x}^2 \, dy+ {y} ^2\, dz$

where C is the line from (1, 0, 0) to (4, 1, 3)

Here

x→ 1⇒ 4

y→  0⇒1

z→  0⇒ 3

Let  t be defined be the range  0≤ t ≤ 1

Then x= 4t+1   :    dx= 4dt

y= t                 ":   dy= dt

z= 3t                : dz= 3dt

Putting the values

$\int\limits^1_0{9t^2} \, 4dt + \int\limits^1_0 {16t^2+1+8t} \, 3dt +\int\limits^1_0 {t^2} \,3dt$

=  [36(1)²- 36(0)²]+ 3 [16(1)² +1+8(1)] - 3 [16(0)² +1+8(0)]  + [3(1)²- 3(0)² ]

= 36 +75-3+3

= 111  

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