VWAP with U-shaped volume

(a) Prove (4A.3). (b) Take $T=1$ and the U-shaped volume density $w(t)=\tfrac12\big(1+3(2t-1)^2\big)$. Check $\int_0^1w=1$, compute $\int_0^1w(t)t\,dt$, and compute $\operatorname{Var}(W_1-\int_0^1wW)$ exactly. (c) With Nifty at 24,000 and intraday volatility 0.7%, give the standard deviation of close minus VWAP in index points for this volume profile and for uniform volume.