Textbook Question
Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.
∫ (from 0 to 1/√3) dt / (t² + 1)^(7/2)
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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 8.7.35
[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.
Find, to two decimal places, the areas of the surfaces generated by revolving the curves in Exercises 35 and 36 about the x-axis.
y = sin x, 0 ≤ x ≤ π
Verified step by step guidance1
Identify the formula for the surface area of a curve revolved around the x-axis. The surface area \( S \) is given by:
\[ S = \int_a^b 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]
For the given curve \( y = \sin x \), determine the interval of integration, which is from \( a = 0 \) to \( b = \pi \).
Compute the derivative of \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \cos x \]
Substitute \( y = \sin x \) and \( \frac{dy}{dx} = \cos x \) into the surface area formula:
\[ S = \int_0^{\pi} 2\pi \sin x \sqrt{1 + \cos^2 x} \, dx \]
Set up the integral for evaluation, and then use a calculator or computer to approximate the value of the integral to two decimal places.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Surface Area of Revolution
The surface area of a solid formed by revolving a curve around an axis is found using an integral formula. For revolution about the x-axis, the formula is S = ∫ 2πy √(1 + (dy/dx)²) dx over the given interval. This calculates the total area of the curved surface generated.
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Example 1: Minimizing Surface Area
Derivative of the Function
To apply the surface area formula, you need the derivative dy/dx of the function y = sin x. The derivative, cos x, measures the slope of the curve at each point and is essential for computing the integrand's square root term, which accounts for the curve's steepness.
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06:30Derivatives of Other Trig Functions
Definite Integration and Numerical Approximation
Evaluating the surface area requires integrating the function from 0 to π. Since the integral may not have a simple closed form, numerical methods or a calculator are used to approximate the value to two decimal places, ensuring an accurate and practical solution.
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05:43Definition of the Definite Integral
Related Practice
Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ x (7x + 5)^(3/2) dx
Textbook Question
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ arcsin(y) dy
Textbook Question
Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.
∫ csc³(√θ) / √θ dθ
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.
∫ (√x / (1 + x³)) dx
Hint: Let u = x^(3/2).
Textbook Question
For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.
∫ √(x² + 2x + 2) / (x² + 2x + 1) dx