Domanda del libro di testo
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ dx / (4 - x²)^(3/2) from 0 to 1
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 8, Problema 8.7.36
[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.
Find, to two decimal places, the areas of the surfaces generated by revolving the curves in Exercises 35 and 36 about the x-axis.
y = x²/4, 0 ≤ x ≤ 2
Guida verificata passo dopo passo1
Identify the formula for the surface area of a solid of revolution about the x-axis:
\[ S = \int_a^b 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]
Given the curve \[ y = \frac{x^2}{4} \], find its derivative with respect to \[ x \]:
\[ \frac{dy}{dx} = \frac{d}{dx} \left( \frac{x^2}{4} \right) = \frac{x}{2} \]
Substitute \[ y \] and \[ \frac{dy}{dx} \] into the surface area formula:
\[ S = \int_0^2 2\pi \left( \frac{x^2}{4} \right) \sqrt{1 + \left( \frac{x}{2} \right)^2} \, dx \]
Simplify the expression inside the square root and the integrand:
\[ S = \int_0^2 2\pi \frac{x^2}{4} \sqrt{1 + \frac{x^2}{4}} \, dx = \int_0^2 \frac{\pi x^2}{2} \sqrt{1 + \frac{x^2}{4}} \, dx \]
Evaluate the integral using an appropriate method (such as substitution or numerical integration) to find the surface area, then round the result to two decimal places.
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Surface Area of Revolution
The surface area of a solid formed by revolving a curve around an axis is found using an integral formula. For revolution about the x-axis, the formula involves integrating 2π times the radius (the y-value) times the arc length element. This concept connects geometry with calculus to measure curved surfaces.
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09:07
Example 1: Minimizing Surface Area
Arc Length Element (ds)
The arc length element ds represents a small segment of the curve and is given by √(1 + (dy/dx)²) dx. It accounts for the curve's slope, ensuring the surface area calculation accurately follows the curve's shape rather than just the x-interval.
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Percorso guidato
06:29Arc Length of Parametric Curves
Definite Integration with Limits
Definite integration calculates the exact accumulated value over an interval, here from x = 0 to x = 2. Applying limits ensures the surface area corresponds precisely to the specified portion of the curve, providing a numerical result to two decimal places.
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Percorso guidato
05:43Definition of the Definite Integral
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