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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 6 - Applications of Integration문제 6.5.29a
6장, 문제 6.5.29a

21–30. {Use of Tech} Arc length by calculator


a. Write and simplify the integral that gives the arc length of the following curves on the given interval. 
y = 1/x, for 1 ≤ x ≤ 10

검증된 단계별 안내
1
Recall the formula for the arc length of a curve given by a function \( y = f(x) \) on the interval \( [a, b] \): \[ L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]
Identify the function and interval: here, \( y = \frac{1}{x} \) and \( x \) ranges from 1 to 10.
Compute the derivative \( \frac{dy}{dx} \) of \( y = \frac{1}{x} \). Use the power rule or rewrite \( y = x^{-1} \) and differentiate.
Square the derivative \( \left(\frac{dy}{dx}\right)^2 \) and add 1 inside the square root to form the integrand: \[ \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \]
Write the integral for the arc length explicitly with the limits 1 to 10: \[ L = \int_1^{10} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \] This integral can then be evaluated using a calculator or numerical methods.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m
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주요 개념

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

Arc Length Formula

The arc length of a curve y = f(x) from x = a to x = b is found using the integral L = ∫_a^b √(1 + (dy/dx)^2) dx. This formula calculates the length of the curve by summing infinitesimal line segments along the curve.
추천 영상:
가이드 코스
06:29
Arc Length of Parametric Curves

Derivative of the Function

To apply the arc length formula, you need the derivative dy/dx of the function y = 1/x. The derivative measures the slope of the curve at each point and is essential for computing the integrand √(1 + (dy/dx)^2).
추천 영상:
06:30
Derivatives of Other Trig Functions

Definite Integral Evaluation

After setting up the integral for arc length, evaluating it over the interval [1, 10] gives the total length. This may require simplification or numerical methods, such as using a calculator, especially when the integral cannot be expressed in elementary functions.
추천 영상:
가이드 코스
05:43
Definition of the Definite Integral
관련 실천
교과서 질문

Volume of a sphere Let R be the region bounded by the upper half of the circle x²+y² = r² and the x-axis. A sphere of radius r is obtained by revolving R about the x-axis.


a. Use the shell method to verify that the volume of a sphere of radius r is 4/3 πr³.

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교과서 질문

13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.


a. Determine when the motion is in the positive direction and when it is in the negative direction. 


v(t) = 50e^−2t on [0, 4]

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교과서 질문

55–58. Marginal cost Consider the following marginal cost functions.


a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units.


C′(x)=200−0.05x

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교과서 질문

Functions from arc length What differentiable functions have an arc length on the interval [a, b] given by the following integrals? Note that the answers are not unique. Give a family of functions that satisfy the conditions.

a. ∫a^b √1+16x⁴ dx

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교과서 질문

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


a. The distance traveled by an object moving along a line is the same as the displacement of the object.

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교과서 질문

Let R be the region in the first quadrant bounded above by the curve y=2−x² and bounded below by the line y=x. Suppose the shell method is used to determine the volume of the solid generated by revolving R about the y-axis.

a. What is the radius of a cylindrical shell at a point x in [0, 2]?

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