Variance of Integral of a real white Gaussian Noise Process
Since $W(t)$ is assumed to be zero-mean, also the RV $Y$ is zero-mean. Hence, the variance of $Y$ is given by
$$\begin{align}\sigma_Y^2&amp;=E\left\{Y^2\right\}\\&amp;=E\left\{\int_{-\infty}^{\infty}W(t_1)\phi(t_1)dt_1\int_{-\infty}^{\infty}W(t_2)\phi(t_2)dt_2\right\}\\&amp;=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\phi(t_1)\phi(t_2)E\big\{W(t_1)W(t_2)\big\}dt_1dt_2\tag{1}\end{align}$$
where $E\big\{W(t_1)W(t_2)\big\}$ is the auto-correlation function $R_W(t_2-t_1)$ of $W(t)$. Now you just have to figure out the expression for $R_W(\tau)$ and solve the integral $(1)$.