Domanda del libro di testo
Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.
y=x^2,y=2−x, and y=0; about the y-axis
79
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 6.5.12
9–20. Arc length calculations Find the arc length of the following curves on the given interval.
y = 3 ln x− x²/24 on [1, 6]
Guida verificata passo dopo passo1
Recall the formula for the arc length of a curve \(y = f(x)\) on the interval \([a, b]\):
\[L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\]
Find the derivative \(\frac{dy}{dx}\) of the given function \(y = 3 \ln x - \frac{x^2}{24}\). Use the derivative rules for logarithmic and polynomial functions:
\[\frac{dy}{dx} = 3 \cdot \frac{1}{x} - \frac{2x}{24}\]
Simplify the derivative expression:
\[\frac{dy}{dx} = \frac{3}{x} - \frac{x}{12}\]
Substitute \(\frac{dy}{dx}\) into the arc length formula under the square root:
\[L = \int_1^6 \sqrt{1 + \left(\frac{3}{x} - \frac{x}{12}\right)^2} \, dx\]
Set up the integral for evaluation. At this point, you can either simplify the expression inside the square root further or use numerical methods or a calculator to approximate the integral value over the interval \([1, 6]\).
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:
3mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
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 sums the lengths of infinitesimal line segments along the curve, providing the total distance traveled along it.
Video consigliato:
Percorso guidato
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 = 3 ln x − x²/24. The derivative measures the slope of the curve at each point, which is essential for calculating the integrand √(1 + (dy/dx)²).
Video consigliato:
06:30Derivatives of Other Trig Functions
Definite Integration
After finding the integrand, you evaluate the definite integral from x = 1 to x = 6. This process sums the continuous values of √(1 + (dy/dx)²) over the interval, yielding the exact arc length of the curve between the specified bounds.
Video consigliato:
Percorso guidato
05:43Definition of the Definite Integral
Pratica correlata
Domanda del libro di testo
9–20. Arc length calculations Find the arc length of the following curves on the given interval.
y = x^3/2 / 3 − x^1/2 on [4, 16]
90
views
Domanda del libro di testo
45–48. Shell and washer methods about other lines Use both the shell method and the washer method to find the volume of the solid that is generated when the region in the first quadrant bounded by y = x²,y=1, and x=0 is revolved about the following lines.
x = -1
110
views
Domanda del libro di testo
Suppose the region bounded by the curve y=f(x) from x=0 to x=4 (see figure) is revolved about the x-axis to form a solid of revolution. Use left, right, and midpoint Riemann sums, with n=4 subintervals of equal length, to estimate the volume of the solid of revolution.
35
views
Domanda del libro di testo
Use the general slicing method to find the volume of the following solids.
The solid whose base is the region bounded by the curves y=x^2 and y=2−x^2, and whose cross sections through the solid perpendicular to the x-axis are squares
91
views
Domanda del libro di testo
35–38. Shell and washer methods Let R be the region bounded by the following curves. Use both the shell method and the washer method to find the volume of the solid generated when R is revolved about the indicated axis.
y = 8,y = 2x+2,x = 0, and x=2; about the y-axis
53
views