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The base of a solid is the region R in the first quadrant enclosed by the parabola y=x² +2, the line y= 6, and the y-axis, as shown below. Cross sections of the solid perpendicular to the y-axis are regular hexagons, whose area formula is A = 3 (² , where is the side length. Find the volume of the solid. R 2

The base of a solid is the region R in the first quadrant enclosed by the parabola y=x² +2, the line y= 6, and the y-axis, as shown below. Cross sections of the solid perpendicular to the y-axis are regular hexagons, whose area formula is A = 3 (² , where is the side length. Find the volume of the solid. R 2

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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The base of a solid is the region \( R \) in the first quadrant enclosed by the parabola \( y = x^2 + 2 \), the line \( y = 6 \), and the y-axis, as shown below. Cross sections of the solid perpendicular to the y-axis are regular hexagons, whose area formula is \( A = \frac{3\sqrt{3}}{2} \ell^2 \), where \( \ell \) is the side length. Find the volume of the solid. The diagram illustrates the region \( R \) in the coordinate plane. It is located in the first quadrant. The boundary of the region comprises: 1. The parabola \( y = x^2 + 2 \). 2. The horizontal line \( y = 6 \). 3. The y-axis. The region \( R \) is shaded blue and lies under the line \( y = 6 \) and above the parabola \( y = x^2 + 2 \), starting at the y-axis and extending to the point where the line and the parabola intersect. The point of intersection on the graph is at approximately the x-coordinate of 2, where the parabola meets the horizontal line. Find the volume of the solid by evaluating the integral of cross-sectional areas perpendicular to the y-axis. Each cross-section is a regular hexagon with the area given by the formula provided.
Transcribed Image Text:The base of a solid is the region \( R \) in the first quadrant enclosed by the parabola \( y = x^2 + 2 \), the line \( y = 6 \), and the y-axis, as shown below. Cross sections of the solid perpendicular to the y-axis are regular hexagons, whose area formula is \( A = \frac{3\sqrt{3}}{2} \ell^2 \), where \( \ell \) is the side length. Find the volume of the solid. The diagram illustrates the region \( R \) in the coordinate plane. It is located in the first quadrant. The boundary of the region comprises: 1. The parabola \( y = x^2 + 2 \). 2. The horizontal line \( y = 6 \). 3. The y-axis. The region \( R \) is shaded blue and lies under the line \( y = 6 \) and above the parabola \( y = x^2 + 2 \), starting at the y-axis and extending to the point where the line and the parabola intersect. The point of intersection on the graph is at approximately the x-coordinate of 2, where the parabola meets the horizontal line. Find the volume of the solid by evaluating the integral of cross-sectional areas perpendicular to the y-axis. Each cross-section is a regular hexagon with the area given by the formula provided.
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