Question

Evulate ∫ (2x^2 + 3y^2)sc, where C is the curve
C
x(t)= at^2, y(t) = 2at, 0≤t≤1

Answer 1

$∫ (2x^2 dx+ 3y^2dy)\\ x(t)= at^2 \implies dx=2atdt\\ y(t) = 2at\implies dy =2adt\\ ∫ (2x^2 dx+ 3y^2dy)=∫_0^12(at^2)^2*2atdt+3(2at)^22adt\\ ∫_0^14a^3t^5dt+∫_0^112a^3t^2dt\\ ∫_0^14a^3t^5dt+∫_0^112a^3t^2dt\\ \frac{2}{3}a^3+4a^3\\ \frac{14a^3}{3}$