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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 6.1.48a
6장, 문제 6.1.48a

Filling a tank A 2000-liter cistern is empty when water begins flowing into it (at t=0 at a rate (in L/min) given by Q′(t) = 3√t, where t is measured in minutes.


a. How much water flows into the cistern in 1 hour?

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1
Identify the given rate of water flow into the cistern as a function of time: \(Q'(t) = 3\sqrt{t}\) liters per minute, where \(t\) is in minutes.
Recognize that \(Q'(t)\) represents the derivative of the volume of water \(Q(t)\) with respect to time, so to find the total volume of water that has flowed in by time \(t\), you need to integrate \(Q'(t)\) over the interval from 0 to \(t\).
Set up the definite integral to find the total volume of water that has flowed into the cistern in 1 hour (which is 60 minutes): \(\displaystyle Q(60) = \int_0^{60} 3\sqrt{t} \, dt\)
Rewrite the integrand \(3\sqrt{t}\) as \(3t^{1/2}\) to make integration straightforward.
Integrate \(3t^{1/2}\) with respect to \(t\) using the power rule for integration: \(\int 3t^{1/2} \, dt = 3 \cdot \frac{t^{3/2}}{\frac{3}{2}} + C = 2t^{3/2} + C\). Then evaluate this antiderivative from 0 to 60 to find the total volume of water that has flowed into the cistern in 1 hour.

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주요 개념

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

Definite Integral

A definite integral calculates the total accumulation of a quantity over an interval. In this problem, integrating the rate function Q′(t) from 0 to 60 minutes gives the total volume of water that has flowed into the cistern during that time.
추천 영상:
가이드 코스
05:43
Definition of the Definite Integral

Rate of Change and Accumulation

The rate function Q′(t) represents how fast water flows into the tank at any time t. Understanding that integrating this rate over time accumulates the total amount of water is essential to solving the problem.
추천 영상:
가이드 코스
04:16
Intro To Related Rates

Units and Time Conversion

Since the rate is given in liters per minute and time is in minutes, it is important to convert the time interval correctly (1 hour = 60 minutes) to ensure the integral limits match the units and yield a meaningful total volume.
추천 영상:
5:10
Evaluate Composite Functions - Values on Unit Circle
관련 실천
교과서 질문

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a. How many barrels are produced in the first 35 days?

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교과서 질문

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d. Determine the position function s(t) using the Fundamental Theorem of Calculus (Theorem 6.1). Check your answer by finding the position function using the antiderivative method.


v(t) = 12t²-30t+12, for 0 ≤ t ≤ 3; s(0)=1

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교과서 질문

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

c. If water flows into a tank at a constant rate (for example, 6 gal/min), the volume of water in the tank increases according to a linear function of time.

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Oscillating growth rates Some species have growth rates that oscillate with an (approximately) constant period P. Consider the growth rate function N'(t) = r+A sin 2πt/P, where A and r are constants with units of individuals/yr, and t is measured in years. A species becomes extinct if its population ever reaches 0 after t=0.


a. Suppose P=10, A=20, and r=0. If the initial population is N(0)=10, does the population ever become extinct? Explain.

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교과서 질문

Work in a gravitational field For large distances from the surface of Earth, the gravitational force is given by F(x) = GMm / (x+R)², where G = 6.7×10^−11 N m²/kg² is the gravitational constant, M = 6×10^24 kg is the mass of Earth, m is the mass of the object in the gravitational field, R = 6.378×10⁶ m is the radius of Earth, and x≥0 is the distance above the surface of Earth (in meters).


a. How much work is required to launch a rocket with a mass of 500 kg in a vertical flight path to a height of 2500 km (from Earth’s surface)?

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교과서 질문

A surface is generated by revolving the line f(x)=2−x, for 0≤x≤2, about the x-axis. Find the area of the resulting surface in the following ways.


a. Using calculus

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