Find the value of the integral $$ \int_{C}(x+y) d x+x^{2} y d y $$ Atong $$ y=x^{2} $$ having $$ (0,0),(3,9) $$ end points (7mks)

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We parameterize the curve $$ y = x^{2} $$ by letting
$$
x = t, \quad y = t^2, \quad \text{with } t \in [0, 3].
$$
Then,
$$
dx = dt \quad \text{and} \quad dy = 2t\ dt.
$$

Substitute into the integral
$$
\int_{C} (x+y)\,dx + x^{2}y\,dy:
$$
$$
\int_{t=0}^{3} \left[(t+t^2)(dt) + t^{2}(t^2)(2t\, dt)\right] = \int_{0}^{3} \left[t+t^2 + 2t^5\right] dt.
$$

We now integrate each term separately:

1. The integral of $$ t $$:
$$
\int_{0}^{3} t\ dt = \left.\frac{t^2}{2}\right|_{0}^{3} = \frac{9}{2}.
$$

2. The integral of $$ t^2 $$:
$$
\int_{0}^{3} t^{2}\ dt = \left.\frac{t^3}{3}\right|_{0}^{3} = \frac{27}{3} = 9.
$$

3. The integral of $$ 2t^5 $$:
$$
\int_{0}^{3} 2t^5\ dt = 2\left.\frac{t^6}{6}\right|_{0}^{3} = \left.\frac{t^6}{3}\right|_{0}^{3} = \frac{3^6}{3} = \frac{729}{3} = 243.
$$

Adding the results, we get:
$$
\frac{9}{2} + 9 + 243 = \frac{9}{2} + \frac{18}{2} + \frac{486}{2} = \frac{9+18+486}{2} = \frac{513}{2}.
$$

Thus, the value of the integral is
$$
\boxed{\frac{513}{2}}.
$$
The value of the integral is $$ \frac{513}{2} $$.

Find the value of the integral

Atong having
end points (7mks)

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Find the value of the integral Atong having end points (7mks)