Evaluate integral _C x ds, where C is a. the straight line segment x = t, y = t/2, from (0, 0) to (12, 6)
b. the parabolic curve x = t, y = 3t^2, from (0, 0) to (2, 12)

a.    

b.  

Step-by-step explanation:

Evaluate integral _C x ds  where C is

a. the straight line segment x = t, y = t/2, from (0, 0) to (12, 6)

i . e

where;

x = t   , y = t/2

the derivative of x with respect to t is:

the derivative of y with respect to t is:

and t varies from 0 to 12.

we all know that:

∴

=

b. the parabolic curve x = t, y = 3t^2, from (0, 0) to (2, 12)

Given that:

x = t  ; y = 3t²

the derivative of  x with respect to t is:

the derivative of y with respect to t is:

Hence; the  integral _C x ds is:

Let consider u to be equal to  1 + 36t²

1 + 36t² = u

Then, the differential of t with respect to u is :

76 tdt = du

The upper limit of the integral is = 1 + 36× 2² = 1 + 36×4= 145

Thus;

it would be:

c.parallelogram.