Question

Evaluate _C∫ (2x² + 3y²)dx, where C is the curve given by x(t)=at² , y(t)=2at, 0≤t≤1.

Answer 1

Let us evaluate $\int_C (2x^2 + 3y^2)dx$, where $C$ is the curve given by $x(t)=at^2 , y(t)=2at, 0≤t≤1.$

$\int_C (2x^2 + 3y^2)dx=\int_0^1 (2(at^2)^2 + 3(2at)^2)d(at^2)=\int_0^1 (2a^2t^4 + 12a^2t^2)2atdt= \int_0^1 (4a^3t^5 + 24a^3t^3)dt=4a^3\int_0^1 (t^5 + 6t^3)dt=4a^3(\frac{t^6}{6}+6\frac{t^4}{4})|_0^1= 4a^3(\frac{1}{6}+\frac{3}{2})=4a^3\frac{10}{6}=\frac{20a^3}{3}.$