To solve the problem of finding the surface area of a curve rotated about the $y$-axis, we use the formula:

$A = 2\pi \int_{a}^{b} x \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$

Given:

$x = 3t^2, \quad y = 2t^3, \quad 0 \leq t \leq 5$

Step 1: Compute $\frac{dx}{dt}$ and $\frac{dy}{dt}$

$\frac{dx}{dt} = \frac{d}{dt}(3t^2) = 6t, \quad \frac{dy}{dt} = \frac{d}{dt}(2t^3) = 6t^2$

Step 2: Substitute into the formula

The surface area formula becomes: $A = 2\pi \int_{0}^{5} (3t^2) \sqrt{(6t)^2 + (6t^2)^2} \, dt$

Simplify the square root: $(6t)^2 = 36t^2, \quad (6t^2)^2 = 36t^4 \quad \Rightarrow \quad (6t)^2 + (6t^2)^2 = 36t^2 + 36t^4 = 36t^2(1 + t^2)$

$\sqrt{(6t)^2 + (6t^2)^2} = \sqrt{36t^2(1 + t^2)} = 6t\sqrt{1 + t^2}$

Thus: $A = 2\pi \int_{0}^{5} (3t^2)(6t\sqrt{1 + t^2}) \, dt$

Step 3: Simplify the integrand

$A = 2\pi \int_{0}^{5} 18t^3\sqrt{1 + t^2} \, dt$

Step 4: Use substitution for integration

Let $u = 1 + t^2$, so $du = 2t \, dt$. When $t = 0, u = 1$. When $t = 5, u = 26$.

The integral becomes: $A = 2\pi \int_{1}^{26} 18 \cdot \frac{u - 1}{2} \, du$

function t = symbols('t') u = 1 + t**2

Define the integrand

integrand = 18 * t3 * sqrt(1 + t2)

Perform the integration

surface_area = 2 * pi * integrate(integrand, (t, 0,The surface area of the surface obtained by rotating the curve around the $y$-axis is:

$A = 2\pi \left(\frac{12}{5} + \frac{11388\sqrt{26}}{5}\right)$

This can also be simplified numerically if needed. Let me know if you want the exact decimal value.

Do you want further explanation, a numerical answer, or details about any step?

Here are related questions to consider:

1. What is the surface area formula for rotation about the $x$-axis?
2. How do parametric equations influence surface area computation?
3. Can we compute surface area without using substitution techniques?
4. How does changing the limits of integration affect the surface area?
5. What are common applications of surface area in engineering and physics?

Tip: Always check if the parameterization of the curve simplifies calculations or introduces complications!