Question 27.5: Evaluate the integral ∫C F(x, y) • ds, where F(x, y) = (3x²y......

The curve C has equation y=2 x^2 \text { for } 2 \leqslant x \leqslant 3. Along this curve we can replace y by 2 x^2 \text {. Note also that } \frac{\mathrm{d} y}{\mathrm{~d} x}=4 x and so we can replace dv with 4x dx. This will produce an integral entirely in terms of the variable x whieh is then integrated between x = 2 and x = 3. Thus

\begin{aligned}=\int_2^3 25 x^2-8 x^3 \mathrm{~d} x\end{aligned}

\begin{aligned} & =\left[\frac{25 x^3}{3}-2 x^4\right]_2^3 \\ & =28.333 \end{aligned}