Problem 44 - Thomas Calculus in SI Units

Question

Find the volume of the solid generated by revolving each region about the \(y\) -axis. The region in the first quadrant bounded on the left by the circle \(x^{2}+y^{2}=3,\) on the right by the line \(x=\sqrt{3},\) and above by the line \(y=\sqrt{3}\)

Step-by-Step Solution

Answer

The volume is \( \pi \sqrt{3} \).

Step 1: Understanding the Region

The region is bounded by the circle \( x^2 + y^2 = 3 \), the vertical line \( x = \sqrt{3} \), and the horizontal line \( y = \sqrt{3} \). All parts are confined to the first quadrant.

Step 2: Identifying the Method

The volume of a solid of revolution can be found using the disk or washer method. Here, we use the washer method because we're revolving around the \( y \)-axis.

Step 3: Setting up the Washer Method

The washer method requires the outer radius, which extends from \( x = 0 \) to the circle, and the inner radius from \( x = 0 \) to the line \( x = \sqrt{3} \). Calculating in terms of \( y \), the outer radius is given by \( x = \sqrt{3 - y^2} \), and the inner radius is constant at \( x = \sqrt{3} \).

Step 4: Establishing Limits

The limits of integration are from \( y = 0 \) to \( y = \sqrt{3} \) because these are the bounds of the region on the \( y \)-axis.

Step 5: Writing the Volume Integral

The volume is given by \[ V = \pi \int_0^{\sqrt{3}} \left( (\sqrt{3})^2 - (\sqrt{3 - y^2})^2 \right) \, dy \] This simplifies to \[ V = \pi \int_0^{\sqrt{3}} \left( 3 - (3 - y^2) \right) \, dy \] which simplifies further to \[ V = \pi \int_0^{\sqrt{3}} y^2 \, dy \]

Step 6: Evaluating the Integral

Compute the integral: \[ \int_0^{\sqrt{3}} y^2 \, dy = \left[ \frac{y^3}{3} \right]_0^{\sqrt{3}} \] This evaluates to \[ \frac{(\sqrt{3})^3}{3} = \frac{3\sqrt{3}}{3} = \sqrt{3} \]

Step 7: Calculating the Volume

Substitute the result of the integral back into the volume formula to get \[ V = \pi \cdot \sqrt{3} \]

Step 8: Finalizing the Answer

The final volume of the solid is \( \pi \sqrt{3} \).

Key Concepts

Washer Method

The washer method is a powerful technique in integral calculus used to find the volume of a solid of revolution. It is particularly useful when the solid has a hollow section, similar to a washer. This method calculates the volume by subtracting the volume of an inner solid from an outer solid.
To use the washer method, we first need to determine two things: the outer radius and the inner radius. These radii are plotted along the axis of revolution which, in this case, is the y-axis.

• The outer radius extends from the axis of revolution to the farthest point of the solid, which is the boundary of the circular region.
• The inner radius is the distance from the axis of revolution to the nearest boundary or hollow part, which is determined by a line or curve inside the region.

Once the radii are determined, the washer cross-section area can be found by taking the difference between the area of the outer circle and the inner circle. The formula for the volume of the solid, integrating along the axis, is:
\[ V = \pi \int_{a}^{b} \left( R(y)^2 - r(y)^2 \right) \, dy \]
where \( R(y) \) and \( r(y) \) are the outer and inner radii as functions of the variable of integration.

Solid of Revolution

A solid of revolution is a three-dimensional object created by rotating a two-dimensional region around an axis. Understanding solids of revolution is crucial for solving these kinds of calculus problems, as it involves visualizing how a shape evolves through rotation.
To generate a solid of revolution, you need:

• An axis of rotation: This can be the x-axis, y-axis, or any other line in the plane of the region.
• A region to rotate: The region is defined by functions or lines that act as boundaries.

By rotating the region around the axis, the solid is formed. The volume of this solid can be difficult to calculate directly, which is where calculus methods like the washer and disk methods come into play. Each method uses integral calculus principles to calculate the volume by summing infinite thin slices of the region's profile as it sweeps around the axis.

Integral Calculus

Integral calculus is a branch of mathematics that focuses on the accumulation of quantities and the areas under and between curves. It is integral (no pun intended) to determining volumes, especially in solving problems involving solids of revolution.
The main idea of integral calculus is to find the sum of infinitely small areas or volumes to obtain a total. This is done through the process of integration, often using definite integrals to find specific numerical values over set intervals.
In the context of solving volume problems, it allows us to determine the space inside a curved shape by using a formula for the integral. By calculating the area under a function's curve and rotating it around an axis, we can use integral calculus to derive the volume of a solid of revolution with methods like the disk or washer method.
Key terms in integral calculus include:

• Integrand: The function to be integrated.
• Limits of Integration: The points that define the starting and stopping points of integration.
• Definite Integral: The result of calculating an integral over a specified interval.

Calculus Problem Solving

Solving calculus problems, particularly those involving volumes of solids of revolution, requires a step-by-step approach to ensure all variables and methods are correctly applied.
Begin by clearly understanding the problem and determining the shape and boundaries of the region in question. Identifying the correct method to calculate the volume is crucial, whether it’s the disk or washer method.
Subsequently, you will need to set up the correct integral that models the volume, taking care to define the outer and inner radii accurately. Using calculus, evaluate the integral, remembering to apply your limits of integration. This involves computations, including substitution and simplification of expressions to find the final volume.

Keys to Success

• Visualize the problem by sketching graphs and diagrams.
• Carefully set up your integral with correct functions and limits.
• Double-check calculations and simplify where necessary.
• Practice regularly to get familiar with different types of problems.