Problem 5 - Thomas Calculus in SI Units

Question

Find the volumes of the solids. The base of a solid is the region between the curve \(y=2 \sqrt{\sin x}\) and the interval \([0, \pi]\) on the \(x\) -axis. The cross-sections perpendicular to the \(x\) -axis are a. equilateral triangles with bases running from the \(x\) -axis to the curve as shown in the accompanying figure. b. squares with bases running from the \(x\) -axis to the curve.

Step-by-Step Solution

Answer

The volume with equilateral triangles is \(2\sqrt{3}\), and with squares is 8.

Step 1: Understand the Problem

We need to find the volume of a solid whose base is the region bound by the curve \(y = 2\sqrt{\sin x}\) and the \(x\)-axis over the interval \([0, \pi]\). The cross-sections perpendicular to the \(x\)-axis form specific shapes: equilateral triangles in part (a) and squares in part (b).

Step 2: Volume with Equilateral Triangle Cross-Sections

For part (a), consider an equilateral triangle with side length \(s = 2\sqrt{\sin x}\), since the length is from the \(x\)-axis to the curve. The area \(A\) of such a triangle is \(A = \frac{\sqrt{3}}{4} s^2\). Therefore, \(A = \frac{\sqrt{3}}{4} (2\sqrt{\sin x})^2 = \sqrt{3} \sin x\). The volume \(V\) of the solid is then obtained by integrating this area over \([0, \pi]\): \[ V = \int_{0}^{\pi} \sqrt{3} \sin x \, dx.\]

Step 3: Calculate the Integral for Equilateral Triangles

Calculate the integral for the equilateral triangle case:\[ V = \sqrt{3} \int_{0}^{\pi} \sin x \, dx.\]The integral of \(\sin x\) is \(-\cos x\), so evaluate it from \(0\) to \(\pi\):\[ V = \sqrt{3} [-\cos x]_{0}^{\pi} = \sqrt{3} [-(-1) - (-1)] = \sqrt{3} (2) = 2\sqrt{3}.\]

Step 4: Volume with Square Cross-Sections

For part (b), the cross-sectional area of a square is \(A = s^2\), where \(s = 2\sqrt{\sin x}\). Thus, \[ A = (2\sqrt{\sin x})^2 = 4\sin x.\]The volume \(V\) of the solid is \[ V = \int_{0}^{\pi} 4\sin x \, dx.\]

Step 5: Calculate the Integral for Squares

Calculate the integral for the square cross-section: \[ V = 4 \int_{0}^{\pi} \sin x \, dx.\]Utilizing the fact that the integral of \(\sin x\) is \(-\cos x\), evaluate:\[ V = 4 [-\cos x]_{0}^{\pi} = 4 [-(-1) - (-1)] = 8.\]

Key Concepts

Integral Calculus

Integral calculus is a fundamental part of calculus that deals with accumulation of quantities. It helps us find values like areas, volumes, and more based on integrals. In this problem, integral calculus is used to determine the volume of a three-dimensional solid. The main idea is to think of the solid as a collection of thin slices, with each slice having a calculable volume.
This method is especially useful when dealing with solids whose cross-sections change continuously along one axis. By slicing the solid and integrating the area of these slices over a specific interval, you can find the total volume.
In this particular problem, for both equilateral triangle and square cross-sections, integration is used to add up the contribution of each thin slice to the total volume of the solid.

Solid of Revolution

A solid of revolution is a 3D object obtained by rotating a 2D shape around an axis. While this problem isn't strictly about a solid of revolution, understanding this concept helps in grasping how solids can be formed from cross-sections.
When a cross-sectional shape, like an equilateral triangle or square, varies continuously along one axis, it can form a solid by stacking these shapes end-to-end in a specific spatial manner.
The base of our solids in this exercise is not formed by a revolution, but the principle of using cross-sectional areas to derive the volume is similar. This technique is a powerful method for visualizing and calculating volumes of various complex solids.

Cross-Sectional Area

Cross-sectional area is essential in understanding how to find the volume of solids. In this problem, we look at the curve's function, which determines the base length of the cross-sectional areas. For equilateral triangles, the area can be calculated via the formula for an equilateral triangle's area using the side length.
Here, the side length is derived from the curve equation giving us the dimensions of the triangle bases and squares. For triangles, each cross-section has an area of \( A = \frac{\sqrt{3}}{4} s^2 \), and for squares, \( A = s^2 \).
These calculated areas are then used in the integral calculus process to sum across the entire interval, resulting in the total volume of each solid in the problem.

Mathematical Problem Solving

Mathematical problem-solving in calculus involves breaking down problems into manageable steps. Each step helps simplify the complex elements, making it easier to tackle the question as a whole.
This problem guides us through a series of logical steps, starting from understanding the problem, calculating the area of each cross-section, then setting up and evaluating the integral to find the volume.
By breaking the original question into smaller parts—like differentiating between different cross-sectional shapes (triangles vs squares) and utilizing known calculus techniques (such as integral evaluation)—we systematically solve the problem. This structured approach is key in tackling calculus problems efficiently.