Question

Write an integral that represents the area of the surface generated by revolving the curve about the \(x\) -axis. Use a graphing utility to approximate the integral. $$ x=\frac{1}{4} t^{2}, \quad y=t+2 \quad 0 \leq t \leq 4 $$

Short answer

You'll use a graphing calculator or other tool to estimate the integral, as the exact numeric answer isn't easily computed by hand.

Step-by-step solution

Write down the surface of revolution formula

Surface area \(A\) of revolution around the x-axis of a curve defined by parameters \(x(t)\) and \(y(t)\) is calculated using the formula: \[ A = \int_a^b 2 \pi y(t) \sqrt{[x'(t)]^2 + [y'(t)]^2} dt \]

Calculate derivatives of \(x(t)\) and \(y(t)\)

For the provided parametric equations: \(x(t) = \frac{1}{4}t^2\) and \(y(t) = t + 2\),Calculate \(x'(t)\) and \(y'(t)\) as the derivatives of the above equations: \[x'(t) = \frac{1}{2}t, \quad y'(t) = 1\]

Substitute into the formula

Substitute \(x'(t)\), \(y'(t)\), and \(y(t)\) into the integral from step 1:\[A = \int_0^4 2 \pi (t + 2) \sqrt{(\frac{1}{2}t)^2 + 1^2} dt \]

Simplify the integral

Simplify the square root and the integral: \[A = 2 \pi \int_0^4 (t + 2) \sqrt{\frac{1}{4}t^2 + 1} dt \]

Use a graphing utility to approximate the integral

Plug the integral into a graphing calculator or other utility to estimate the area.

Key concepts

Parametric Equations

When exploring complex curves in mathematics, parametric equations provide a powerful tool. These equations express a set of quantities as explicit functions of one or more independent parameters. Usually denoted as t in the context of curves, this parameter can be thought of as time, which traces the path of the curve as it varies.

In the exercise at hand, the curve is defined by two parametric equations, where \( x = \frac{1}{4} t^2 \) and \( y = t + 2 \). Here, t acts as our independent variable, and as it changes, the values of x and y adjust accordingly, plotting a curve in the xy-plane as 't' ranges from 0 to 4. This approach allows for the description of more complex curves that might be difficult to express in a standard y=f(x) form.

Understanding the relationship between the independent parameter and the curve's coordinates is pivotal in calculus, especially when calculating lengths, areas, and volumes related to the curve.

Surface Integral

The surface integral is a fundamental concept when dealing with surfaces in calculus, especially those generated by revolving a curve around an axis. This integral takes into account not only the shape and size of the curve but also the manner in which it is swept around the axis to create a surface.

To calculate a surface area of revolution, as in our exercise, we use a specialized form of the surface integral that incorporates the circumference of a circle, \(2\text{\textpi}r\), with \(r\) being the distance from the axis of revolution. The formula provided as the surface of revolution formula, \[ A = \text{\textint}_a^b 2 \text{\textpi} y(t) \text{\textradical \text{{[}}\text{{x}}'(t)\text{{]}}^2 + \text{{[}}\text{{y}}'(t)\text{{]}}^2} dt \], utilizes the parametric equations to express the radius and account for the changing shape of the curve.

Executing the surface integral requires the calculation of the derivatives of the parametric equations and substituting those into the integral. This process gives a precise mathematical formula for the surface area that can be evaluated, often with the aid of a graphing utility to handle the complexity of the integration.

Calculus Graphing Techniques

Visualization is a key component in understanding calculus problems. Calculus graphing techniques are not only useful for illustrating concepts but also serve as practical tools for approximating mathematical solutions, like the one needed in our exercise.

A graphing utility can plot the curve defined by the parametric equations, showing us the relationship between the variables x, y, and the parameter t. Once the integral form of the surface area is derived, as shown in the exercise, a graphing utility becomes indispensable. It allows us to approximate the value of the integral, given that exact solutions are often unobtainable for more complex integrals.

Graphing the function under the integral sign can also provide a visual understanding of the region being integrated, the behavior of the curve, and its revolution around the axis. This visual insight can be particularly beneficial when comprehending the nature of the surface generated and the expected results of the integration.