Question

Surface area using technology Consider the following curves on the given intervals. a. Write the integral that gives the area of the surface generated when the curve is revolved about the \(x\) -axis. b. Use a calculator or software to approximate the surface area. $$y=\tan x \text { on }\left[0, \frac{\pi}{4}\right]$$

Short answer

Answer: The approximate surface area is \(2.82843\) square units.

Step-by-step solution

Find the derivative of the function

Before we can use the surface area formula, we need to find the derivative \(\frac{dy}{dx}\) of the curve \(y = \tan x\). We can find the derivative using the basic derivative rules. $$\frac{dy}{dx}=\frac{d}{dx}\left(\tan (x)\right) = \sec^2(x)$$ Now that we have the derivative, we can proceed to the next step.

Setup the integral for surface area

To find the surface area generated when the curve is revolved about the x-axis, we'll use the following formula: $$S = 2\pi\int_{a}^{b}y\sqrt{1+\left(\frac{dy}{dx}\right)^2} dx$$ For our problem, \(y = \tan x\), \(\frac{dy}{dx} = \sec^2(x)\), and the interval we're considering is \(\left[0, \frac{\pi}{4}\right]\). Substituting these values into the formula, we get: $$S = 2\pi\int_{0}^{\frac{\pi}{4}}\tan(x)\sqrt{1+\sec^4(x)} dx$$ Now, we'll use technology to approximate the surface area.

Approximate the integral using technology

To approximate the value of the integral, we can use a calculator or software that can handle definite integrals. Using a calculator or software like Wolfram Alpha, we find that: $$S \approx 2\pi\int_{0}^{\frac{\pi}{4}}\tan(x)\sqrt{1+\sec^4(x)} dx \approx 2.82843$$ Therefore, the surface area of the solid generated when the curve \(y = \tan x\) on the interval \(\left[0, \frac{\pi}{4}\right]\) is revolved about the x-axis is approximately \(2.82843\) square units.

Key concepts

Integration

Integration is the mathematical process we use to add up small quantities to find a total or sum. In calculus, integration is used to determine quantities like area, volume, and in this case, surface area. By integrating a function, you can find the total cumulative effect of the function over an interval. For example, to find the surface area of a shape created by revolving a curve around an axis, we need to set up an integral. This method accumulates the surface contributions from each infinitesimally small piece of the curve to find the total surface area. This comprehensive addition is what makes integration such a powerful tool in calculus.

Derivative

The derivative of a function gives us the rate at which the function is changing at any given point. It forms the foundational idea for both differential calculus and practical applications relating to change. When you're dealing with problems that involve surface area, like when we revolve a curve around an axis, the derivative helps to adjust for the curve's slope. This creates a more accurate model of the surface. For example, in our exercise, we found that the derivative of the function \( y = \tan(x) \) is \( \frac{dy}{dx} = \sec^2(x) \). This derivation is necessary before setting up the integral that will eventually solve the problem of the surface area for us.

Parametric Equations

Parametric equations allow us to describe a curve using a parameter, typically denoted by \( t \). These equations are very helpful when dealing with curves that are not easily represented with standard Cartesian equations. While our exercise doesn't use parametric equations directly, understanding them is beneficial for comprehending how curves can be expressed and manipulated mathematically. They allow seamless transformation and simplification of complex curves, making them amenable to analysis like integration and differentiation, similar to what we need to compute surface areas.

Calculator/Software Usage

In modern mathematics, calculator and software usage have become integral in performing complex calculations that would otherwise be too time-consuming or difficult by hand. For integrals, especially those with complex expressions like \( S = 2\pi\int_{0}^{\frac{\pi}{4}}\tan(x)\sqrt{1+\sec^4(x)} \: dx \), technology can approximate values efficiently. Tools like graphing calculators, Wolfram Alpha, or TI-Nspire can compute the definite integral accurately to find exact, numerical solutions. They provide significant aid in ensuring calculations are precise and mistakes are minimized, which is critical in mathematical problems.

Definite Integral

A definite integral calculates the net "accumulation" of a quantity over a specific interval \( [a, b] \). Unlike an indefinite integral, which gives a family of solutions, the definite integral provides a concrete numerical value. In our exercise, the definite integral \( \int_{0}^{\frac{\pi}{4}}\tan(x)\sqrt{1+\sec^4(x)} \: dx \) gives us the exact surface area of the solid created by revolving the curve \( y = \tan(x) \) around the x-axis within the interval \( [0, \frac{\pi}{4}] \). Understanding this integral is crucial for calculating real-life applications where exact measurements are necessary.