Domanda del libro di testo
Area and volume The region R is bounded by the curves x = y²+2,y=x−4, and y=0 (see figure).
a. Write a single integral that gives the area of R.
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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
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 6, Problema 6.R.58
58–61. Arc length Find the length of the following curves.
y = 2x+4 on [−2,2] (Use calculus.)
Guida verificata passo dopo passo1
Recall the formula for the arc length of a curve defined 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 the interval: here, \(y = 2x + 4\) and the interval is \([-2, 2]\).
Compute the derivative of \(y\) with respect to \(x\):
\[\frac{dy}{dx} = \frac{d}{dx}(2x + 4)\]
Substitute the derivative into the arc length formula:
\[L = \int_{-2}^{2} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\]
Evaluate the integral to find the length of the curve over the given interval.
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Arc Length Formula
The arc length of a curve y = f(x) from x = a to x = b is found using the integral formula L = ∫_a^b √(1 + (dy/dx)^2) dx. This formula calculates the length by summing infinitesimal line segments along the curve.
Video consigliato:
Percorso guidato
06:29
Arc Length of Parametric Curves
Derivative of a Function
The derivative dy/dx represents the slope of the curve at any point x. For the arc length formula, the derivative is squared and added to 1 inside the square root to account for both horizontal and vertical changes along the curve.
Video consigliato:
06:30Derivatives of Other Trig Functions
Definite Integration
Definite integration evaluates the integral over a specific interval [a, b], providing a numerical value for the arc length. It sums the continuous contributions of small segments between the given bounds.
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Percorso guidato
05:43Definition of the Definite Integral
Pratica correlata
Domanda del libro di testo
Area and volume The region R is bounded by the curves x = y²+2,y=x−4, and y=0 (see figure).
b. Write a single integral that gives the volume of the solid generated when R is revolved about the x-axis.
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Domanda del libro di testo
43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.
The region bounded by the graphs of y = 2x,y = 6−x, and y = 0 is revolved about the line y = −2 and the line x = −2. Find the volumes of the resulting solids. Which one is greater?
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Domanda del libro di testo
43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.
The region bounded by the graph of y = 4−x² and the x-axis on the interval [−2,2] is revolved about the line x = −2. What is the volume of the solid that is generated?
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Domanda del libro di testo
35-38. Area and volume Let R be the region in the first quadrant bounded by the graph of
Find the area of the region R.
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Domanda del libro di testo
Variable gravity At Earth’s surface, the acceleration due to gravity is approximately g=9.8 m/s² (with local variations). However, the acceleration decreases with distance from the surface according to Newton’s law of gravitation. At a distance of y meters from Earth’s surface, the acceleration is given by a(y) = - g / (1+y/R)², where R=6.4×10⁶ m is the radius of Earth.
f. Graph ymax as a function of v0. What is the maximum height when v0=500 m/s,1500 m/s, and 5 km/s?
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