Find the area of the surface generated when the given curve is revolved about the given axis.


x=√12y−y^2, for 2≤y≤10; about the y-axis

Verified step by step guidance

1. Step 1: Recall the formula for the surface area of a curve revolved about the y-axis. The formula is: $A\; =\; 2\pi \; \int [a,b]\; x\; \surd (1\; +\; (dx/dy)^2)\; dy$, where x is expressed as a function of y, and the integral is evaluated over the given interval [a, b].
2. Step 2: Identify the given curve and interval. The curve is $x\; =\; \surd (12y\; -\; y^2)$, and the interval is $2\; \le \; y\; \le \; 10$. Substitute this expression for x into the formula.
3. Step 3: Compute $dx/dy$. Differentiate $x\; =\; \surd (12y\; -\; y^2)$ with respect to y using the chain rule. This gives $dx/dy\; =\; (1/2)(12y\; -\; y^2)^(-1/2)\; *\; (12\; -\; 2y)$. Simplify this derivative.
4. Step 4: Substitute $dx/dy$ into the formula for surface area. The integrand becomes $x\; \surd (1\; +\; (dx/dy)^2)$, where $x\; =\; \surd (12y\; -\; y^2)$ and $dx/dy$ is the derivative computed in Step 3.
5. Step 5: Set up the definite integral for the surface area. The integral is $A\; =\; 2\pi \; \int [2,10]\; \surd (12y\; -\; y^2)\; \surd (1\; +\; ((1/2)(12y\; -\; y^2)^(-1/2)\; *\; (12\; -\; 2y))^2)\; dy$. Simplify the integrand as much as possible, and then proceed to evaluate the integral using appropriate techniques such as substitution or numerical methods.

Key Concepts

Surface Area of Revolution

The surface area of revolution is calculated by rotating a curve around an axis. The formula for the surface area generated by revolving a curve y = f(x) about the y-axis is given by S = 2π ∫ x * √(1 + (dy/dx)²) dy, where the integral is evaluated over the specified interval. This concept is essential for determining the area of the surface created by the rotation of the curve.

Parametric Equations

In this problem, the curve is given in terms of y, which can be expressed as a function of y (x = √(12y - y²)). Understanding how to manipulate and interpret parametric equations is crucial for finding the derivatives needed in the surface area formula. This involves differentiating x with respect to y to find dy/dx, which is necessary for the surface area calculation.

Integration

Integration is a fundamental concept in calculus used to find areas, volumes, and other quantities. In this context, it involves evaluating the integral of the surface area formula over the specified limits (from y = 2 to y = 10). Mastery of definite integrals is essential for accurately computing the total surface area generated by the revolution of the curve.