Domanda del libro di testo
{Use of Tech} Decreasing velocity A projectile is fired upward, and its velocity in m/s is given by v(t) = 200e^−t/10, for t≥0.
a. Graph the velocity function, for t≥0.
230
views
Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
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.52
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?
Guida verificata passo dopo passo1
First, visualize the region bounded by the curve \(y = 4 - x^{2}\) and the x-axis on the interval \([-2, 2]\). This region lies above the x-axis and below the parabola between \(x = -2\) and \(x = 2\).
Since the solid is generated by revolving this region about the vertical line \(x = -2\), consider using the shell method because the axis of rotation is vertical and the slices perpendicular to the x-axis will be easier to handle.
Set up a representative cylindrical shell at a position \(x\) between \(-2\) and \(2\). The height of the shell is given by the function value \(h(x) = 4 - x^{2}\), and the radius of the shell is the horizontal distance from \(x\) to the line \(x = -2\), which is \(r(x) = x - (-2) = x + 2\).
The volume of each thin shell is given by the formula \(dV = 2\pi \times \text{radius} \times \text{height} \times \text{thickness} = 2\pi (x + 2)(4 - x^{2}) \, dx\).
To find the total volume, integrate the expression for \(dV\) over the interval \(x = -2\) to \(x = 2\): \(V = \int_{-2}^{2} 2\pi (x + 2)(4 - x^{2}) \, dx\). This integral can then be evaluated to find the volume.
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
7mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Volume of Solids of Revolution
This concept involves finding the volume of a 3D solid formed by rotating a 2D region around a line (axis of revolution). The volume is computed by integrating cross-sectional areas perpendicular to the axis, using methods like disks, washers, or shells depending on the axis and shape.
Video consigliato:
04:48
Finding Volume Using Disks
Shell Method
The shell method calculates volume by summing cylindrical shells formed by revolving vertical or horizontal slices around an axis. Each shell's volume is 2π(radius)(height)(thickness), making it useful when the axis of rotation is parallel to the slices and not the coordinate axis.
Video consigliato:
07:33Euler's Method
Setting up the Integral with Correct Limits and Radius
Accurately determining the radius and height of slices relative to the axis of rotation is crucial. For rotation about x = -2, the radius is the horizontal distance from x to -2, and the height is given by the function y = 4 - x². Proper limits correspond to the interval of the region.
Video consigliato:
Percorso guidato
11:11Improper Integrals: Infinite Intervals
Pratica correlata
Domanda del libro di testo
27–33. Multiple regions The regions R₁,R₂, and R₃ (see figure) are formed by the graphs of y = 2√x,y = 3−x,and x=3.
Find the area of each of the regions R₁,R₂, and R₃.
58
views
Domanda del libro di testo
58–61. Arc length Find the length of the following curves.
y = 2x+4 on [−2,2] (Use calculus.)
51
views
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.
152
views
Domanda del libro di testo
Comparing volumes Let R be the region bounded by y=1/x^p and the x-axis on the interval [1, a], where p>0 and a>1 (see figure). Let Vₓ and Vᵧ be the volumes of the solids generated when R is revolved about the x- and y-axes, respectively.
c. Find a general expression for Vₓ in terms of a and p. Note that p=1/2 is a special case. What is Vₓ when p=1/2?
60
views
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.
63
views