교과서 질문
Find the area of the region described in the following exercises.
The region bounded by y=x^2−2x+1 and y=5x−9
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Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
6장, 문제 6.4.4
Look again at the region R in Figure 6.38 (p. 439). Explain why it would be difficult to use the washer method to find the volume of the solid of revolution that results when R is revolved about the y-axis.
검증된 단계별 안내1
Identify the region R and the axis of revolution, which is the y-axis in this case.
Recall that the washer method involves slicing the solid perpendicular to the axis of revolution, creating washers (disks with holes) whose volume can be integrated.
When revolving around the y-axis, the slices should be horizontal (parallel to the x-axis) to form washers with radii expressed as functions of y.
If the region R is bounded by curves that are easier to express as functions of x rather than y, it becomes difficult to describe the inner and outer radii of the washers in terms of y, complicating the integral setup.
Therefore, the difficulty arises because the boundaries of R may not be easily invertible or expressible as functions of y, making the washer method cumbersome for this axis of revolution.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Washer Method for Volumes of Revolution
The washer method involves slicing a solid perpendicular to the axis of revolution, creating washers (disks with holes). It requires expressing the outer and inner radii as functions of the variable of integration, typically aligned with the axis of revolution.
추천 영상:
04:48
Finding Volume Using Disks
Axis of Revolution and Variable of Integration
When revolving a region about the y-axis, the variable of integration is usually y. To use the washer method effectively, the boundaries of the region must be easily expressed as functions of y, which can be challenging if the region is defined primarily in terms of x.
추천 영상:
06:30Disk Method Using y-Axis
Complexity of Expressing Radii in Terms of y
If the region R is bounded by curves given as functions of x, rewriting these boundaries as functions of y may be difficult or impossible to do explicitly. This complicates finding the inner and outer radii for washers, making the washer method less practical.
추천 영상:
4:22Expand & Condense Log Expressions
관련 실천
교과서 질문
Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.
y=4−x^2,x=2, and y=4; about the y-axis
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교과서 질문
9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.
x = x³ ,y = 1, and x = 0; about the x-axis
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교과서 질문
29–36. Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2).
a(t) = −32; v(0)=50; s(0)=0
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교과서 질문
Express the area of the shaded region in Exercise 5 as the sum of two integrals with respect to y. Do not evaluate the integrals.
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교과서 질문
Find the area of the region described in the following exercises.
The region bounded by y=√x, y=2x−15, and y=0
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