Question

What is the arc length of the curve given by `r(t)=(4t,3t-6)` in the interval `t in [0,7]`?

Answer 1

35

Explanation:

The Arc Length of `f(t)=(x(t), y(t) )` is given by:

`L = int_a^bsqrt((dx/dt)^2 + (dy/dt)^2)dt`

Let `x(t)=4t => dx/dt=4`
And, `y(t)=3t-6=>dy/dt=3`

So,

`L = int_0^7sqrt((4)^2 + (3)^2)dt`
`:. L = int_0^7sqrt(16+9)dt`
`:. L = int_0^7sqrt(25)dt`
`:. L = 5int_0^7 dt`
`:. L = 5[t]_0^7`
`:. L = 5(7-0)`
`:. L = 35`