Textbook Question
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?
Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 6, Problem 6.6.27a
Consider the following curves on the given intervals.
a. Write the integral that gives the area of the surface generated when the curve is revolved about the given axis.
y=tan x , for 0≤x≤π/4; about the x-axis
Verified step by step guidance1
Identify the formula for the surface area generated by revolving a curve about the x-axis. The surface area \( S \) is given by the integral \( S = \int_a^b 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \), where \( y = f(x) \).
Given the curve \( y = \tan x \) on the interval \( 0 \leq x \leq \frac{\pi}{4} \), substitute \( y = \tan x \) into the formula.
Compute the derivative \( \frac{dy}{dx} = \sec^2 x \).
Substitute \( y = \tan x \) and \( \frac{dy}{dx} = \sec^2 x \) into the surface area integral to get \( S = \int_0^{\frac{\pi}{4}} 2\pi \tan x \sqrt{1 + (\sec^2 x)^2} \, dx \).
Simplify the expression inside the square root as much as possible before setting up the integral for evaluation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Surface Area of Revolution
The surface area generated by revolving a curve around an axis is found using an integral formula that accounts for the curve's length and distance from the axis. For a function y = f(x) revolved about the x-axis, the surface area is given by the integral of 2π times the radius (y) times the arc length element.
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Example 1: Minimizing Surface Area
Arc Length Element (ds)
The differential arc length element ds represents a small segment of the curve and is calculated as ds = √(1 + (dy/dx)^2) dx. This accounts for both horizontal and vertical changes along the curve, ensuring the surface area integral accurately measures the curved surface.
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06:29Arc Length of Parametric Curves
Trigonometric Function Differentiation
To set up the integral, you need the derivative of y = tan x, which is sec^2 x. Understanding how to differentiate trigonometric functions is essential to find dy/dx, which is used in the arc length formula for the surface area calculation.
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6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
Determine whether the following statements are true and give an explanation or counterexample.
a. If the curve y=f(x) on the interval [a, b] is revolved about the y-axis, the area of the surface generated is ∫f(b)f(a) 2πf(y)√1+f′(y)^2 dy.
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. When using the shell method, the axis of the cylindrical shells is parallel to the axis of revolution.
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Textbook Question
Bike race Theo and Sasha start at the same place on a straight road, riding bikes with the following velocities (measured in mi/hr). Assume t is measured in hours.
Theo: vT(t)=10, for t≥0
Sasha: vS(t)=15t, for 0≤t≤1, and vS(t)=15, for t>1
a. Graph the velocity function for both riders.
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Textbook Question
Explain the steps required to find the length of a curve x = g(y) between y=c and y=d.
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Textbook Question
Calculating work for different springs Calculate the work required to stretch the following springs 1.25 m from their equilibrium positions. Assume Hooke’s law is obeyed.
a. A spring that requires 100 J of work to be stretched 0.5 m from its equilibrium position