Textbook Question
Use any method to evaluate the integrals in Exercises 55–66.
∫ x² √(1 - x²) dx
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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
Chapter 8, Problem 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.
Verified step by step guidance1
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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5mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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Guided course
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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07:33Euler'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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06:35Derivatives of Other Inverse Trigonometric Functions
Related Practice
Textbook Question
Expand the quotients in Exercises 1–8 by partial fractions.
(5x - 7) / (x² - 3x + 2)
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Textbook Question
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.
Textbook Question
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
Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ tan^(-1)(x) / x² dx
Textbook Question
Centroid: Find the centroid of the region bounded by the x-axis, the curve y = csc x, and the lines x = π/6, x = 5π/6.
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