Problem 1 - Thomas Calculus

Question

Find the volumes of the solids. The solid lies between planes perpendicular to the \(x\) -axis at \(x=0\) and \(x=4 .\) The cross-sections perpendicular to the axis on the interval \(0 \leq x \leq 4\) are squares whose diagonals run from the parabola \(y=-\sqrt{x}\) to the parabola \(y=\sqrt{x}\) .

Step-by-Step Solution

Answer

The volume of the solid is 16 cubic units.

Step 1: Identify Cross-Section Formula

Since the cross-sections of the solid are squares, we start by understanding that the length of the diagonal of each cross-section square is given by the distance between the two parabolas, which is \(d = \sqrt{x} - (-\sqrt{x}) = 2\sqrt{x}\. Since this is the diagonal, we can find the side of the square (\s\) using the relation for the diagonal in terms of the side length of a square: \d = s\sqrt{2}\. Therefore, \[s = \frac{d}{\sqrt{2}} = \sqrt{2x}.\]

Step 2: Calculate Area of Cross-Section

The area of a square is the square of its side length. Therefore, the area of each cross-section is given by \[A(x) = (\sqrt{2x})^2 = 2x.\]

Step 3: Integrate to Find Volume

To find the volume of the solid, integrate the area function along the x-axis from 0 to 4. The integral is set up as follows: \[V = \int_0^4 2x \; dx.\]

Step 4: Evaluate the Integral

Evaluate the integral: \[V = \int_0^4 2x \; dx = \left[ x^2 \right]_0^4 = 4^2 - 0^2 = 16.\] Thus, the volume of the solid is 16 cubic units.

Key Concepts

Calculating Integrals

Integrals are a fundamental concept in calculus, used to determine things like area, volume, and accumulation of quantities. An integral essentially adds up an infinite number of infinitely small quantities to find totals. For volume calculations, especially when dealing with solids of revolution or solids with known cross-section shapes, integrals help us sum up the volume of these tiny cross-sectional slices.
The integral in our problem aims to find the volume of a solid with cross-sections that vary along the x-axis between two points. In this case, from 0 to 4. The function we integrate, called the integrand, represents the area of the cross-section at each point x. Integrating from x=0 to x=4 gives us the total volume of the solid.

Cross-Section Method

The cross-section method is a technique for finding the volume of a solid where the cross-sectional area can be expressed as a function of their position along an axis. This method is particularly useful when the solid is too complex for direct volume computation.

• First, identify the shape of cross-sections perpendicular to a specified axis (here, squares).
• Determine the dimensions or area of a typical cross-section as a function of position along the axis (here, it’s a square determined by the distance between the parabolas).
• Set up and evaluate an integral of this area function over the required interval (from x=0 to x=4).

By dividing the solid into these cross-sections, we take advantage of their simpler geometric properties to calculate volume.

Area of Cross-Sections

In our problem, the cross-sections are squares, which are fundamentally defined by the relation between their side length and diagonal. The diagonal of a square can be related to its side length by the formula: diagonal = side\( \times \sqrt{2} \). Here, the diagonal spans from one parabola to the opposite one, giving us a diagonal length of \(2\sqrt{x}\) determined by the distance \( \sqrt{x} - (-\sqrt{x}) \).
Using this diagonal, we find the side length \(s\) as \( \frac{2\sqrt{x}}{\sqrt{2}} = \sqrt{2x} \). Hence, the area \(A(x)\) of each square cross-section is given by \((\sqrt{2x})^2 = 2x\).
This area function is what we integrate to find the total volume of the solid.

Parabolas

Parabolas are a type of U-shaped curve, defined by a quadratic equation like \(y = x^2\). In this problem, the parabolas are defined by \(y = \sqrt{x}\) and \(y = -\sqrt{x}\), which mirror one another across the x-axis.
These particular parabolas extend through both positive and negative y coordinates for each value of x, creating a symmetric pattern. The distance between them across any vertical line is the length of the diagonal of the square cross-sections we've discussed.
Understanding these parabolas is crucial since they determine the bounds and dimensions for the cross-sectional squares, an essential part of getting the correct volume for the solid.