Question

What is the net area between `f(x) = xsinx` and the x-axis over `x in [0, 3pi ]`?

Answer 1

Net Area `= 3 pi = 9.42478`

Explanation:

Area `=int_0^pi x*sin x` `dx` +`int_pi^(2pi) x*sin x` `dx` +`int_(2pi)^(3pi) x*sin x` `dx`

Use integration by parts:

`int u` `dv` = `uv - int v` `du`

Let `u=x`,

`dv` = `sin x` `dx`

`v= - cos x`

`du`= `dx`

`int x*sin x` `dx`=`-x* cos x-int (-cos x)` `dx`

`= -x* cos x+int cos x` `dx`

`=-x* cos x+sin x`

Apply the limits `[0, pi], and [pi, 2pi], and [2pi, 3pi]`

the areas are `pi and -3pi and 5pi`

so that the Net Area = `pi+(-3pi)+5pi= 3pi`