Given the Euclidean vector space
\left(\mathcal P_{\leq 2}, \langle \ , \ \rangle \right) with
\displaystyle \langle p, q \rangle = \int_{-1}^{1} p(x)q(x) \; dx
and the polynomial {\color{red}p} \; : [-1,1] \to \mathbb R
with {\color{red}p(x) =Bx^2 +C}.
Determine the length L = \|{\color{red}p} \|.
L =
L
It is
\displaystyle L = \sqrt{\langle {\color{red}p},{\color{red}p}\rangle}
.
By definition and with the given polynomial we have
\displaystyle
\int_{-1}^{1} {\color{red}\left(Bx^2 +C\right)^2} \; dx =
\int_{-1}^{1} \left(B*Bx^4 + 2*C*Bx^2 + C*C\right) \; dx =
negParens(fractionReduce(B*B,5)) x^5 +
fractionReduce(2*C*B,3)x^3 + C*Cx \biggl|_{-1}^{1} =
fractionReduce(2*B*B*3+ 20*C*B+30*C*C,15).
We take the square root to get
L = \sqrt{fractionReduce(2*B*B*3+ 20*C*B+30*C*C,15)}.