Find the average value of the function...

Question

Find the average value of the function $$F(x) = \int_x^1 \sin(t^2) \, dt$$ on $[0,1]$.

I know the average value of a function $f(x)$ on $[a,b]$ is $f_\text{avg}=\dfrac1{b-a} \int_a^b f(x) \, dx$, but I don't know how to apply that to this question... The function loos like the Fresnel integral? But that doesn't quite help me either.

Answer 1

$$ \begin{align} \int_0^1\int_x^1\sin\left(t^2\right)\,\mathrm{d}t\,\mathrm{d}x &=\int_0^1\int_0^t\sin\left(t^2\right)\,\mathrm{d}x\,\mathrm{d}t\\ &=\int_0^1\sin\left(t^2\right)\,t\,\mathrm{d}t\\ &=\frac12\int_0^1\sin\left(t^2\right)\,\mathrm{d}t^2\\ &=\frac12\left[-\cos\left(t^2\right)\right]_0^1\\[3pt] &=\frac{1-\cos(1)}2 \end{align} $$

Answer 2

You can use $\displaystyle f_\text{avg} = \frac{1}{1-0} \int_0^1 \int_x^1 \sin(t^2) \, dt\,dx = \int_0^1 \int_0^t \sin(t^2) \, dx\,dt$