Question

Astroid Find the area of the surface formed by revolving the portion in the first quadrant of the graph of \(x^{2 / 3}+y^{2 / 3}=4\) \(0 \leq y \leq 8\) about the \(y\) -axis.

Short answer

The surface area is given by the integral \(SA = \int_0^8 2 \pi (4 - y^{2/3})^{3/2} \sqrt{1 + y^{-2/3}(4 - y^{2/3})} dy\). Computing this definite integral will yield the answer.

Step-by-step solution

Getting x in terms of y

We isolate x in the equation \(x^{2 / 3} + y^{2 / 3} = 4\). This gives us \(x = (4 - y^{2/3})^{3/2}\).

Finding ds in terms of dy

We know that \(ds = \sqrt{1 + (dx/dy)^2} dy\). Thus, we need to find \(dx/dy\). Differentiating \(x = (4 - y^{2/3})^{3/2}\), we get \(dx/dy = -y^{-1/3}(4 - y^{2/3})^{1/2}\). Hence, \(ds = \sqrt{1 + y^{-2/3}(4 - y^{2/3})} dy\).

Computing the integral

Now, using \(x\) and \(ds\) expressions obtained in steps 1 and 2, the surface area, \(SA\), is given by \(SA = \int_0^8 2 \pi (4 - y^{2/3})^{3/2} \sqrt{1 + y^{-2/3}(4 - y^{2/3})} dy\).

Computing the definite integral

This integral is outside of the standard integrals and might need to be evaluated numerically or using a mathematical software.

Key concepts

Astroid Curve

Imagine tracing the path of a point on the circumference of a circle as it rolls inside another circle. This intriguing pattern created is known as the astroid curve. It's a particular type of hypocycloid, a curve traced by a point on a small circle that rolls within a larger one.

The astroid is characterized by its star-shaped structure and has the parametric equations:
\( x(t) = a \cos^3(t) \) and \( y(t) = a \sin^3(t) \), where \( a \) is the radius of the larger circle and \( t \) is the parameter, usually the angle.

In calculus, this equation can be represented by the implicit form \(x^{2/3} + y^{2/3} = a^{2/3}\), where the astroid curves in all four quadrants. However, in our exercise, we only consider the portion in the first quadrant for revolving about the y-axis.

Parametric Equations Derivatives

When curves are expressed in parametric form, such as the astroid curve, calculating derivatives requires a slightly different approach. Parametric equations separate the variables, and each is expressed in terms of a third variable, typically time (\(t\)).

In our scenario, \(x\) and \(y\) are both functions of \(t\). The derivative of \(x(t)\) with respect to \(t\) is written as \(\frac{dx}{dt}\), and similarly the derivative of \(y(t)\) with respect to \(t\) is \(\frac{dy}{dt}\). To find the slope of the curve at a point, we use the formula \(\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}\), which is crucial when computing the length element \(ds\) for the surface area of the curve.

Definite Integrals

A definite integral is essentially a calculation of the area under a curve between two points on the x-axis. It provides a way to measure the accumulation of quantities, which could be areas, volumes, or, as in our example, surface areas.

The standard notation of a definite integral is \(\int_{a}^{b} f(x) dx\), where \(f(x)\) is the function under consideration, and \(a\) and \(b\) are the limits of integration. In the exercise, we deal with \(SA = \int_0^8 2 \pi (4 - y^{2/3})^{3/2} \sqrt{1 + y^{-2/3}(4 - y^{2/3})} dy\), where \(SA\) is the surface area, and the integral computes this area as the shape is revolved around the y-axis.

Numerical Integration Methods

Sometimes, an integral cannot be solved analytically due to its complexity. This is where numerical integration methods come to the rescue, providing approximate solutions to definite integrals. Common numerical techniques include the Trapezoidal Rule, Simpson’s Rule, and Monte Carlo integration.

These methods approach the area under the curve by breaking it into smaller shapes, like trapezoids or parabolas, summing their areas to get a total approximation. Computational tools and software often employ these methods to determine integrals that are otherwise intractable by hand. In our exercise, since we are faced with a complex integral outside the scope of elementary functions, numerical integration would be the practical approach to approximate the desired surface area.