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Ch. 6 - Applications of Definite Integrals
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 6 - Applications of Definite Integrals문제 6.PE.10b
6장, 문제 6.PE.10b

Volumes
Find the volume of the solid generated by revolving the region bounded by the parabola y² = 4x and the line y = x about
b. the y-axis

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1
First, identify the region bounded by the curves. The parabola is given by \(y^{2} = 4x\), which can be rewritten as \(x = \frac{y^{2}}{4}\). The line is \(y = x\), or equivalently \(x = y\). Find the points of intersection by setting \(\frac{y^{2}}{4} = y\).
Solve the equation \(\frac{y^{2}}{4} = y\) to find the intersection points. Multiply both sides by 4 to get \(y^{2} = 4y\), then rearrange to \(y^{2} - 4y = 0\), and factor as \(y(y - 4) = 0\). So, \(y = 0\) or \(y = 4\) are the limits of integration.
Since the solid is generated by revolving the region about the y-axis, use the method of cylindrical shells. The formula for the volume is \(V = \int_{a}^{b} 2\pi (\text{radius})(\text{height}) \, dy\), where the radius is the distance from the y-axis and the height is the horizontal distance between the curves.
Determine the radius and height for the shell at a given \(y\). The radius is the distance from the y-axis, which is \(x\). The height is the difference between the rightmost and leftmost \(x\) values for the region at that \(y\). Here, the right boundary is \(x = y\) (from the line) and the left boundary is \(x = \frac{y^{2}}{4}\) (from the parabola). So, height = \(y - \frac{y^{2}}{4}\).
Set up the integral for the volume: \(V = \int_{0}^{4} 2\pi y \left(y - \frac{y^{2}}{4}\right) dy\). This integral represents the volume of the solid generated by revolving the region about the y-axis.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
10m

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Understanding the Region Bounded by Curves

To find the volume of a solid of revolution, first identify the region bounded by the given curves. Here, the parabola y² = 4x and the line y = x intersect, defining the limits of integration. Understanding their intersection points and the area enclosed is essential for setting up the integral correctly.
추천 영상:
05:06
Finding Area When Bounds Are Not Given

Method of Cylindrical Shells for Volume

When revolving a region around the y-axis, the cylindrical shells method is often used. This involves integrating the volume of thin cylindrical shells with radius equal to the x-value, height given by the difference in y-values, and thickness dx. This method simplifies volume calculation when the axis of revolution is vertical.
추천 영상:
04:48
Finding Volume Using Disks

Setting Up and Evaluating Definite Integrals

After determining the method and limits, set up the definite integral representing the volume. This requires expressing variables appropriately, integrating with respect to x or y, and carefully evaluating the integral to find the exact volume. Mastery of integral calculus techniques is crucial for accurate computation.
추천 영상:
가이드 코스
05:43
Definition of the Definite Integral
관련 실천
교과서 질문

Volumes

Find the volume of the solid generated by revolving the “triangular” region bounded by the curve y = 4/x³ and the lines x = 1 and y = 1/2 about

c. the line x = 2

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교과서 질문

Volumes

Find the volume of the solid generated by revolving the region bounded on the left by the parabola x = y² + 1 and on the right by the line x = 5 about

c. the line x = 5

34
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교과서 질문

Volumes

Find the volume of the solid generated by revolving the region between the x-axis and curve y = x² ―2x about

d. the line y = 2

59
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교과서 질문

Areas of Surfaces of Revolution

In Exercises 23–26, find the areas of the surfaces generated by revolving the curves about the given axes.

_______

y = √4y ― y² , 1 ≤ y ≤ 2 ; y-axis

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