Question

The force applied against a moving object travelling on a linear path is given by `F(x)=3x^2+x`. How much work would it take to move the object over `x in [1,3 ]`?

Answer 1

here,work will be done against the force to move the object,

so,work done `dW =F dx` (as,angle between `F` and `dx` is zero,as moving along a linear pathway)

so, `dW =(3x^2 +x)dx`

or, `dW =3x^2 dx +x dx`

so, `int _0^W = 3 int_1^3 x^2dx + int_1^3 x dx`

so, `W=[x^3]_1^3 + 1/2 [x^2]_1^3=30J`

Answer 2

`W = 30` J

Explanation:

Simply integrate using the bounds provided:
`W = \int_1^3 F(x)dx`

`= \int_1^3 (3x^2+x) dx`

`=3\int_1^3x^2 dx + \int_1^3xdx`

`= (x^3+1/2x^2)_(x=1)^(x=3)`

`=27+9/2-(1+1/2)`

`=30` J