Domanda del libro di testo
Moment about y-axis:
A thin plate of constant density δ = 1 occupies the region enclosed by the curve
y = 36/(2x + 3) and the line x = 3 in the first quadrant. Find the moment of the plate about the y-axis.
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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 8.7.32
The length of one arch of the curve y = sin x is given by
L = ∫(from 0 to π) √(1 + cos²(x)) dx.
Estimate L by Simpson's Rule with n = 8.
Guida verificata passo dopo passo1
Identify the function to integrate: here, the integrand is \( f(x) = \sqrt{1 + \cos^{2}(x)} \).
Determine the interval and number of subintervals: the integral is from \( 0 \) to \( \pi \), and \( n = 8 \) subintervals means the step size is \( h = \frac{\pi - 0}{8} = \frac{\pi}{8} \).
Calculate the values of \( f(x) \) at the points \( x_0 = 0, x_1 = h, x_2 = 2h, \ldots, x_8 = 8h = \pi \). For each \( x_i \), compute \( f(x_i) = \sqrt{1 + \cos^{2}(x_i)} \).
Apply Simpson's Rule formula: \[ L \approx \frac{h}{3} \left[ f(x_0) + 4 \sum_{i=1,3,5,7} f(x_i) + 2 \sum_{i=2,4,6} f(x_i) + f(x_8) \right] \].
Sum the weighted function values according to Simpson's Rule and multiply by \( \frac{h}{3} \) to estimate the length \( L \).
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Arc Length of a Curve
The arc length of a curve y = f(x) from x = a to x = b is given by the integral L = ∫_a^b √(1 + (dy/dx)²) dx. This formula measures the distance along the curve by summing infinitesimal line segments, accounting for changes in both x and y.
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Percorso guidato
06:29
Arc Length of Parametric Curves
Simpson's Rule
Simpson's Rule is a numerical method to approximate definite integrals by dividing the interval into an even number of subintervals and fitting parabolas through the points. It provides an accurate estimate by weighting function values at endpoints, midpoints, and interior points.
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5:50Power Rules
Partitioning the Interval (n = 8)
Choosing n = 8 means dividing the interval [0, π] into 8 equal subintervals for applying Simpson's Rule. This partitioning determines the step size and the points at which the integrand is evaluated, affecting the accuracy of the numerical approximation.
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08:44Interval of Convergence
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