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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 8 - Techniques of Integration문제 8.7.35
8장, 문제 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

검증된 단계별 안내
1
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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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.
추천 영상:
09:07
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.
추천 영상:
06:30
Derivatives 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.
추천 영상:
05:43
Definition of the Definite Integral
관련 실천
교과서 질문

[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 = x²/4, 0 ≤ x ≤ 2

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교과서 질문

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

∫ from 2 to ∞ of (dx / √(x² - 1))

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교과서 질문

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

∫ from 1 to ∞ of ((1 / (e^x - 2^x)) dx)

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교과서 질문

For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.

∫ √(x² + 2x + 2) / (x² + 2x + 1) dx

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교과서 질문

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ √(9 - w²) dw / w²

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교과서 질문

Evaluate the integrals in Exercises 23–32.

∫₀^(π/6) √(1 + sin(x)) dx

(Hint: Multiply by √((1 - sin(x)) / (1 - sin(x))))

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