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Chapter 8: Problem 43

Concern the region bounded by \(y=x^{2}\) \(y=1,\) and the \(y\) -axis, for \(x \geq 0 .\) Find the volume of the following solids. The solid whose base is the region and whose crosssections perpendicular to the \(x\) -axis are squares.

Short Answer

Expert verified
The volume of the solid is \( \frac{8}{15} \).

Step by step solution

01

Understanding the Problem

We are to find the volume of a solid whose base lies in the region bounded by the curve \( y = x^2 \), the line \( y = 1 \), and the \( y \)-axis, with \( x \geq 0 \). The cross-sections perpendicular to the \( x \)-axis are squares.
02

Identify the Boundaries

The region of interest is bounded by the parabola \( y = x^2 \), the line \( y = 1 \), and the \( y \)-axis. For \( y \) between 0 and 1, \( x \) ranges from 0 to \( \sqrt{y} \).
03

Determine the Cross-section

For each \( x \), the square cross-section perpendicular to the \( x \)-axis has a side length equal to the length running from \( y = x^2 \) to \( y = 1 \). This length is \( 1 - x^2 \).
04

Calculate the Volume

The area of each square cross-section is \( (1 - x^2)^2 \). To find the volume, integrate this area with respect to \( x \) from \( x = 0 \) to \( x = 1 \) (where \( x \) ranges between \( y = x^2 = 0 \) and \( y = 1 \)). The volume \( V \) is given by \( V = \int_{0}^{1} (1 - x^2)^2 \; dx \).
05

Evaluate the Integral

Expand \( (1 - x^2)^2 = 1 - 2x^2 + x^4 \) and integrate term-by-term:\[\int_{0}^{1} (1 - 2x^2 + x^4) \; dx = \left[ x - \frac{2}{3}x^3 + \frac{1}{5}x^5 \right]_{0}^{1} = \left[ 1 - \frac{2}{3} + \frac{1}{5} \right] = \frac{8}{15}.\]
06

Conclusion

The volume of the solid is \( \frac{8}{15} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integration
Integration is a mathematical tool that allows us to calculate the accumulation of quantities, such as areas, volumes, or any continuous element distributed over a range. In the case of volumes of solids, integration helps in summing up infinitely small cross-sectional areas along a specified boundary.

In this problem, integration is used to calculate the volume of a solid with square cross-sections. The function that defines our cross-sections is based on the curve formulas within the given region. By integrating the area function of these cross-sections over the designated interval (here from 0 to 1), we find the total volume of the solid.
  • First, express the area of each cross-section as a function of a variable (usually x).
  • Integrate this area over the boundaries that define the entire solid.
  • Calculation of this integral gives the volume of the solid.
Square Cross-sections
Square cross-sections mean that when we slice through the solid perpendicularly, each slice resembles a square. Each slice has equal side lengths, defined by the region鈥檚 constraints or boundaries

For our problem, the square cross-sections of the solid lie perpendicular to the x-axis. To determine the side length of these squares, one must consider the range defined by the equations. The side length is given by the vertical distance from the curve \( y = x^2 \) to the horizontal line \( y = 1 \).
  • The side length of each square, perpendicular to the x-axis, is \( 1 - x^2 \).
  • The area of each cross-section, being a square, is \((1 - x^2)^2\).
This description provides the function of the area of the squares to use in our integration process to find the volume.
Boundaries of Regions
Boundaries of a region define the limits within which the solid exists. These are essential to understand the extent of integration and the shape of the cross-sections.

Here, the solid is bounded by:
  • The parabola \( y = x^2 \).
  • The horizontal line \( y = 1 \).
  • The y-axis, which implies \( x \geq 0 \).
Because the highest y value within these boundaries is 1, and the lowest is determined by the parabola \( y = x^2 \), the intersection is used to understand the x-range (from 0 to 1 for this problem). These boundaries provide the limits for integration and the dimensions necessary to compute the volume of the solid.

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Most popular questions from this chapter

Sales of Version 6.0 of a computer software package start out high and decrease exponentially. At time \(t,\) in years, the sales are \(s(t)=50 e^{-t}\) thousands of dollars per year. After two years, Version 7.0 of the software is released and replaces Version \(6.0 .\) Assuming that all income from software sales is immediately invested in government bonds which pay interest at a \(6 \%\) rate compounded continuously, calculate the total value of sales of Version 6.0 over the two-year period.

Explain what is wrong with the statement. The arc length of the curve \(y=\sin x\) from \(x=0\) to \(x=\pi / 4\) is \(\int_{0}^{\pi / 4} \sqrt{1+\sin ^{2} x} d x\).

Find the area between the two spirals \(r=\theta\) and \(r=2 \theta\) for \(0 \leq \theta \leq 2 \pi\)

Explain what is wrong with the statement. The solid obtained by rotating the region bounded by the curves \(y=2 x\) and \(y=3 x\) between \(x=0\) and \(x=5\) around the \(x\) -axis has volume \(\int_{0}^{5} \pi(3 x-2 x)^{2} d x\).

Find the are length of the curve \(r=\theta^{2}\) from \(\theta=0\) to \(\theta=2 \pi\)
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