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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 8 - Techniques of IntegrationProblema 8.7.37
Capitolo 8, Problema 8.7.37

[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.
Use numerical integration to estimate the value of
arcsin(0.6) = ∫ (from 0 to 0.6) dx / √(1 - x²).
For reference, arcsin(0.6) = 0.64350 to five decimal places.

Guida verificata passo dopo passo
1
Recognize that the problem asks to estimate \(\arcsin(0.6)\) using the integral definition: \(\arcsin(0.6) = \int_0^{0.6} \frac{dx}{\sqrt{1 - x^2}}\).
Choose a numerical integration method such as the Trapezoidal Rule, Simpson's Rule, or a numerical integration function on a calculator or computer to approximate the integral.
Divide the interval \([0, 0.6]\) into \(n\) subintervals of equal width \(\Delta x = \frac{0.6 - 0}{n} = \frac{0.6}{n}\), where \(n\) is a positive integer you select based on desired accuracy.
Calculate the function values \(f(x) = \frac{1}{\sqrt{1 - x^2}}\) at the endpoints and at each subinterval point \(x_i = 0 + i \Delta x\) for \(i = 0, 1, 2, ..., n\).
Apply the chosen numerical integration formula using these function values to approximate the integral, which will give an estimate for \(\arcsin(0.6)\).

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Definite Integral

A definite integral calculates the accumulated area under a curve between two limits. It is represented as ∫ from a to b of a function f(x) dx, giving a numerical value. In this problem, the integral from 0 to 0.6 of 1/√(1 - x²) dx represents arcsin(0.6).
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Percorso guidato
05:43
Definition of the Definite Integral

Numerical Integration Methods

Numerical integration approximates the value of definite integrals when an exact solution is difficult. Common methods include the Trapezoidal Rule, Simpson’s Rule, and numerical algorithms implemented in calculators or computers. These methods estimate the integral by summing areas of simple shapes under the curve.
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07:33
Euler's Method

Inverse Trigonometric Functions and Their Integral Representations

The arcsine function, arcsin(x), is the inverse of sine restricted to [-π/2, π/2]. It can be expressed as an integral: arcsin(x) = ∫ from 0 to x of 1/√(1 - t²) dt. Understanding this integral form connects the geometric meaning of arcsin with calculus techniques.
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Derivatives of Other Inverse Trigonometric Functions
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Average value

In a mass-spring-dashpot system like the one in Exercise 65, the mass's position at time t is

y = 4e^(-t)(sin(t) - cos(t)), t ≥ 0.

Find the average value of y over the interval 0 ≤ t ≤ 2π.

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Use reduction formulas to evaluate the integrals in Exercises 41–50.

∫ 2 sin^2(t) sec^4(t) dt

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Evaluate ∫ x³ √(1 - x²) dx using:

c. A trigonometric substitution.

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The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫ e^(z + eᶻ) dz

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Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.

∫ tan^(-1)(x) / x² dx

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The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫ (x² / (x² + 1)) dx

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