Problem 5 - Calculus: Early Transcendentals

Question

Use integration to find the volume of the solid obtained by revolving the region bounded by \(x+y=2\) and the \(x\) - and \(y\) -axes around the \(x\) -axis.

Step-by-Step Solution

Answer

The volume of the solid is \(\frac{8\pi}{3}\).

Step 1: Express y in terms of x

We start by manipulating the equation of the line \(x + y = 2\) to solve for \(y\). Rearrange it to get \(y = 2 - x\). This gives us the expression for \(y\) in terms of \(x\).

Step 2: Set up the integral for the volume

The volume \(V\) of the solid of revolution can be found using the formula for the volume of a solid of revolution around the \(x\)-axis, which is \( V = \pi \int_a^b f(x)^2 \, dx \). Here, \(y = 2 - x\) is revolved around the \(x\)-axis. The limits of integration are from \(x=0\) to \(x=2\), the points where the line intersects the axes.

Step 3: Write down the definite integral

Substitute \(f(x) = 2 - x\) into the volume formula to get: \[ V = \pi \int_0^2 (2 - x)^2 \, dx \].

Step 4: Expand the integrand

Expand the square in the integrand: \((2 - x)^2 = 4 - 4x + x^2\). Now our integral is: \[ V = \pi \int_0^2 (4 - 4x + x^2) \, dx \].

Step 5: Integrate

Calculate the integral by finding the antiderivative:\( \int (4 - 4x + x^2) \, dx = [4x - 2x^2 + \frac{x^3}{3}] \). Evaluate this from \(x = 0\) to \(x = 2\).

Step 6: Evaluate the definite integral

Substitute the limits into the antiderivative:\[ \left(4(2) - 2(2)^2 + \frac{(2)^3}{3}\right) - \left(4(0) - 2(0)^2 + \frac{(0)^3}{3}\right) \]. Simplifying, we get:\[ (8 - 8 + \frac{8}{3}) - 0 = \frac{8}{3} \].

Step 7: Calculate the volume

Multiply the result by \(\pi\) to get the final volume:\[ V = \pi \cdot \frac{8}{3} = \frac{8\pi}{3} \].

Key Concepts

Definite Integral

The definite integral is a fundamental concept in calculus that provides the accumulated value or area under a curve between two points on the x-axis.
In this exercise, the definite integral helps quantify the volume of a solid formed by revolving a region around the x-axis.

• It involves finding the antiderivative (or indefinite integral) of a given function.
• Then, evaluating this antiderivative at the upper and lower limits of integration.
• The result gives the net area – or in this case, volume – accumulated between these points.

Understanding definite integrals is key to linking the rates of change in functions to quantities like area and volume.
This process transforms the original problem into one involving straightforward calculations.

Solid of Revolution

A solid of revolution is formed when a region in a plane is revolved around a line, typically an axis.
This creates a three-dimensional object whose volume can be calculated using integration.

• In our exercise, the region bounded by the line \(x + y = 2\) and the axes, forms the desired shape.
• By revolving this region around the x-axis, we obtain a solid whose cross-sections parallel to the x-y plane are circular discs.
• The formula used is: \( V = \pi \int_a^b [f(x)]^2 \, dx \), which adds up the volume of infinitesimally thin discs.

Recognizing the setup of a solid of revolution helps in visualizing the geometry and applying appropriate techniques to find the volume accurately.

Calculus Problem Solving

Solving calculus problems usually involves a clear strategy, breaking down complex questions into manageable components.
In this situation, you first express one variable in terms of another to use in the volume formula.

• Identify the relevant boundary equations and express one dependent variable in terms of another. Here, it’s solved as \( y = 2 - x \).
• Setup the integral, integrating the square of the function, from the given limits that represent bounds of the region.
• Simplify the function inside the integral to make the work with subsequent steps easier.

By following a logical sequence, calculus becomes approachable, and problems like finding volumes of solids of revolution are easier to understand and solve.

Integration Techniques

Integration techniques are tools that help to perform integration, particularly when dealing with complex functions.
Here are some techniques relevant to solving the exercise:

• **Expanding Quadratics**: Expanding the function \((2 - x)^2\) simplifies the integral by turning it into a polynomial, \(4 - 4x + x^2\).
• **Finding Antiderivatives**: This involves determining a function whose derivative is the given function. For \(4 - 4x + x^2\), the antiderivative is \(4x - 2x^2 + \frac{x^3}{3}\).
• **Evaluating Definite Integrals**: After finding the antiderivative, substitute the upper and lower limits and compute the difference.

Mastering these techniques enhances your ability to tackle a variety of calculus problems, providing a toolkit for capturing both areas and volumes effectively.