Textbook Question
18. Finding volume (Continuation of Exercise 17.) Find the volume of the solid generated by revolving the region R about:
a. the y-axis.
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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.AAE.3
Evaluate the integrals in Exercises 1–6.
∫ x arcsin x dx
Verified step by step guidance1
Identify the integral to solve: \(\int x \arcsin x \, dx\).
Use integration by parts, where you let \(u = \arcsin x\) and \(dv = x \, dx\).
Compute \(du\) by differentiating \(u\): \(du = \frac{1}{\sqrt{1 - x^2}} \, dx\), and compute \(v\) by integrating \(dv\): \(v = \frac{x^2}{2}\).
Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\), which gives \(\int x \arcsin x \, dx = \frac{x^2}{2} \arcsin x - \int \frac{x^2}{2} \cdot \frac{1}{\sqrt{1 - x^2}} \, dx\).
Simplify the remaining integral \(\int \frac{x^2}{2 \sqrt{1 - x^2}} \, dx\) and consider substitution methods to evaluate it.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration by Parts
Integration by parts is a technique used to integrate products of functions. It is based on the product rule for differentiation and follows the formula ∫u dv = uv - ∫v du. Choosing u and dv wisely simplifies the integral, especially when one function becomes simpler upon differentiation.
Recommended video:
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Integration by Parts for Definite Integrals
Inverse Trigonometric Functions
Inverse trigonometric functions, like arcsin(x), are the inverses of the standard trigonometric functions. Understanding their derivatives and integrals is essential, as arcsin(x) has a derivative of 1/√(1 - x²), which often appears in integration problems involving these functions.
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06:35Derivatives of Other Inverse Trigonometric Functions
Algebraic Manipulation in Integration
Algebraic manipulation involves rewriting integrals into simpler or more recognizable forms. This can include substitution, simplifying expressions, or breaking integrals into parts, which helps in applying integration techniques effectively and solving complex integrals step-by-step.
Recommended video:
06:18Integration by Parts for Definite Integrals
Related Practice
Textbook Question
Centroid of a region
Find the centroid of the region in the plane enclosed by the curves y = ±(1 − x²)^(-1/2) and the lines x = 0 and x = 1.
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Textbook Question
Finding volume
The region in the first quadrant that is enclosed by the x-axis and the curve y = 3x√(1 − x) is revolved about the y-axis to generate a solid. Find the volume of the solid.
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Textbook Question
Evaluate the integrals in Exercises 1–6.
∫ dt / (t - √(1 - t²))
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Textbook Question
Finding volume
The region in the first quadrant enclosed by the coordinate axes, the curve y = e^x, and the line x = 1 is revolved about the y-axis to generate a solid. Find the volume of the solid.
Textbook Question
Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.
∫(from π/2 to 2π/3) cos θ dθ / (sin θ cos θ + sin θ)
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