IGNOU BMTC 102 2026 - Evaluate , where C is the curve given by

Question

Evaluate $\displaystyle \int_C (2x^2 + 3y^2) dx$ , where C is the curve given by

$$x(t) = at^2, y(t) = 2at, 0 \leq t \leq 1.$

09 Jan 2026

Answer :

Word Count : 219

We are asked to evaluate the line integral [ \int_C (2x^2 + 3y^2) , dx ] along the curve (x(t) = at^2, , y(t) = 2at, , 0 \le t \le 1). First, compute (dx) in terms of (t): [ x = at^2 \implies \frac{dx}{dt} = 2at \implies dx = 2at , dt ] Next, express (2x^2 + 3y^2) in terms of (t): [ 2x^2 + 3y^2 = 2(at^2)^2 + 3(2at)^2 = 2a^2 t^4 + 3 \cdot 4 a^2 t^2 =

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