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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Key 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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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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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
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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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