Problem 199 - Calculus Volume 1

Question

For the following exercises, find the surface area of the volume generated when the following curves revolve around the \(y\) -axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. $$y=x^{2} \text { from } x=0 \text { to } x=2$$

Step-by-Step Solution

Answer

The surface area is approximately 15.7799 square units.

Step 1: Understand the Problem

We need to find the surface area of the solid formed when the curve \( y = x^2 \) is revolved around the \( y \)-axis from \( x = 0 \) to \( x = 2 \). This involves setting up the integral for surface area calculation.

Step 2: Set up the Surface Area Formula

The formula for the surface area \( A \) of a solid of revolution about the \( y \)-axis is given by: \[ A = 2\pi \int_{a}^{b} x \sqrt{1 + \left( \frac{dx}{dy} \right)^2} \ dy \] Since \( y = x^2 \), we first need to express \( x \) in terms of \( y \).

Step 3: Express \( x \) in terms of \( y \)

Since \( y = x^2 \), we can write \( x \) as a function of \( y \): \( x = \sqrt{y} \). This will be used to transform the integral limits and differentiate accordingly.

Step 4: Determine \( \frac{dx}{dy} \)

Differentiate \( x = \sqrt{y} \) with respect to \( y \). Using the power rule, we get: \( \frac{dx}{dy} = \frac{1}{2\sqrt{y}} \).

Step 5: Transform the Integral of Surface Area

Plug \( x = \sqrt{y} \) and \( \frac{dx}{dy} = \frac{1}{2\sqrt{y}} \) into the integral formula. The surface area integral becomes: \[ A = 2\pi \int_{0}^{4} \sqrt{y} \sqrt{1 + \left( \frac{1}{2\sqrt{y}} \right)^2} \ dy \] where \( y \) ranges from 0 to 4 (since \( y = x^2 \)).

Step 6: Simplify the Integral Expression

Simplify the expression inside the square root of the integral: \( 1 + (\frac{1}{2\sqrt{y}})^2 = 1 + \frac{1}{4y} = \frac{4y + 1}{4y} \). Therefore, the integral becomes: \[ A = 2\pi \int_{0}^{4} \frac{y}{\sqrt{y}} \sqrt{\frac{4y+1}{4y}} \ dy \].

Step 7: Evaluate Using a Calculator

This integral does not have a straightforward antiderivative, so we use a calculator to approximate its value as a numeric solution. The calculator gives: \( A \approx 15.7799 \).

Step 8: Conclude the Result

The surface area of the solid formed by revolving \( y = x^2 \) from \( x = 0 \) to \( x = 2 \) around the \( y \)-axis is approximately \( 15.7799 \) square units.

Key Concepts

Solid of Revolution

When we talk about a solid of revolution, we are discussing a 3D shape created by rotating a 2D curve around an axis. Imagine taking a flat curve and spinning it around a line; the resulting shape resembles an object like a vase or a bell. This is what's known as a solid of revolution. In our exercise, the 2D curve is given by the equation \(y = x^2\), and it is spun around the \(y\)-axis. This process transforms the curve into a 3-dimensional object whose surface area we want to calculate. By understanding how a curve can create a solid through revolution, we develop tools to calculate properties like volume and surface area.

Integral Calculation

Integral calculation is a vital technique in finding the area under a curve and, in our case, determining the surface area of a solid of revolution. The key is setting up an integral that aligns with the problem's parameters. Here, we use the formula for the surface area of a solid revolving around the \(y\)-axis:

• \( A = 2\pi \int_{a}^{b} x \sqrt{1 + \left( \frac{dx}{dy} \right)^2} \ dy \)

This formula allows us to take an infinite number of tiny slices of the rotated curve and addup their surface areas to find the total surface. Although some integrals can be solved exactly, others require numerical approximations when they don’t have a simple antiderivative. A calculator often helps in such scenarios to give an approximate numerical solution.

Curve Revolution

Curve revolution is the concept of rotating a curve around an axis to create a solid. It's important to identify the axis of rotation and ensure the curve is expressed as needed. In this case, the rotation happens around the \(y\)-axis, and we transform the curve equation from \(y = x^2\) to express \(x\) as \(x = \sqrt{y}\). This helps adjust limits and variables that fit the chosen axis of rotation. Knowing how to properly set up the curve's form is crucial to applying integral calculus correctly and achieving the correct surface area result.

Differentiation

Differentiation is a fundamental technique in calculus that helps us determine the rate at which quantities change. In finding the surface area of a solid of revolution, differentiation is used to find \( \frac{dx}{dy} \). After expressing \(x\) in terms of \(y\) as \(x = \sqrt{y}\), we differentiate this equation to get \( \frac{dx}{dy} = \frac{1}{2\sqrt{y}} \). This derivative is crucial because it is a component of the surface area integral formula. Knowing how to differentiate functions accurately is important in many applications, particularly when integrating to find certain properties of solids.