Domanda del libro di testo
76. Different Substitutions
b. Show that ∫(1/√(x - x²)) dx = 2 sin⁻¹√x + C using substitution u = √x
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Ch. 8 - Integration Techniques
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 8, Problema 8.8.67a
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
67. Let f(x) = √(x³ + 1).
a. Find a Midpoint Rule approximation to ∫[1 to 6] √(x³ + 1) dx using n = 50 subintervals.
Guida verificata passo dopo passo1
Identify the integral to approximate: \(\int_{1}^{6} \sqrt{x^{3} + 1} \, dx\) and note that we will use the Midpoint Rule with \(n = 50\) subintervals.
Calculate the width of each subinterval using the formula \(\Delta x = \frac{b - a}{n}\), where \(a = 1\) and \(b = 6\). So, \(\Delta x = \frac{6 - 1}{50}\).
Determine the midpoints of each subinterval. For the \(i\)-th subinterval, the midpoint $x_i^*$ is given by \(x_i^* = a + \left(i - \frac{1}{2}\right) \Delta x\) for \(i = 1, 2, \ldots, 50\).
Evaluate the function \(f(x) = \sqrt{x^{3} + 1}\) at each midpoint $x_i^*$ to get \(f(x_i^*)\).
Form the Midpoint Rule approximation by summing the function values at the midpoints multiplied by the subinterval width: \(M_n = \Delta x \sum_{i=1}^{50} f(x_i^*)\).
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Midpoint Rule for Numerical Integration
The Midpoint Rule approximates the definite integral of a function by dividing the interval into equal subintervals and using the function's value at each subinterval's midpoint to estimate the area. This method improves accuracy over using endpoints and is especially useful when the function is complex or lacks an elementary antiderivative.
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Percorso guidato
05:56
Additional Rules for Indefinite Integrals
Partitioning the Interval into Subintervals
To apply the Midpoint Rule, the interval [a, b] is divided into n equal subintervals, each of width Δx = (b - a)/n. The midpoints of these subintervals are then used to evaluate the function, which helps in approximating the integral by summing the areas of rectangles centered at these midpoints.
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08:44Interval of Convergence
Error Estimation in Numerical Integration
Error estimation provides a bound on the difference between the exact integral and its numerical approximation. Theorem 8.1 typically gives an error bound for the Midpoint Rule involving the second derivative of the function, helping to assess the accuracy of the approximation and determine if the chosen number of subintervals is sufficient.
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Percorso guidato
04:57Determining Error and Relative Error
Pratica correlata
Domanda del libro di testo
63. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. If m is a positive integer, then ∫[0 to π] sin^m(x) dx = 0.
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Domanda del libro di testo
85. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. More than one integration method can be used to evaluate ∫ (1 / (1 - x²)) dx.
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Domanda del libro di testo
Computing areas On the interval [0,2], the graphs of f(x)=x²/3 and g(x)=x²(9−x²)^(-1/2) have similar shapes.
a. Find the area of the region bounded by the graph of f and the x-axis on the interval [0,2].
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Domanda del libro di testo
63. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. If m is a positive integer, then ∫[0 to π] cos^(2m+1)(x) dx = 0.
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Domanda del libro di testo
Gamma function The gamma function is defined by Γ(p) = ∫ from 0 to ∞ of x^(p-1) e^(-x) dx, for p not equal to zero or a negative integer.
b. Use the substitution x = u² and the fact that ∫ from 0 to ∞ of e^(-u²) du = √(π/2) to show that Γ(1/2) = √π.
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