Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₂⁴ dt / [t√(t² − 4)]
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 8.6.60
Moment about y-axis:
A thin plate of constant density δ = 1 occupies the region enclosed by the curve
y = 36/(2x + 3) and the line x = 3 in the first quadrant. Find the moment of the plate about the y-axis.
Verified step by step guidance1
Identify the region and the density: The plate has constant density \( \delta = 1 \), and it occupies the region bounded by the curve \( y = \frac{36}{2x + 3} \), the vertical line \( x = 3 \), and the coordinate axes in the first quadrant (where \( x \geq 0 \) and \( y \geq 0 \)).
Set up the expression for the moment about the y-axis: The moment about the y-axis, \( M_y \), for a lamina with density \( \delta \) is given by the integral \( M_y = \int x \cdot \delta \, dA \), where \( dA \) is the differential area element.
Express the differential area element \( dA \) in terms of \( dx \) and \( dy \): Since the region is bounded by \( y = \frac{36}{2x + 3} \), for each \( x \) between 0 and 3, the vertical slice has height \( y \). So, \( dA = y \, dx = \frac{36}{2x + 3} \, dx \).
Write the integral for the moment about the y-axis: Substitute \( dA \) and \( \delta = 1 \) into the moment formula to get \( M_y = \int_0^3 x \cdot 1 \cdot \frac{36}{2x + 3} \, dx = \int_0^3 \frac{36x}{2x + 3} \, dx \).
Evaluate the integral: To solve \( \int_0^3 \frac{36x}{2x + 3} \, dx \), consider using substitution or algebraic manipulation (such as dividing numerator and denominator or rewriting the integrand) to simplify the integral before integrating.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Moment about the y-axis
The moment about the y-axis measures how the mass of a plate is distributed relative to the y-axis. It is calculated by integrating the product of the x-coordinate and the mass element over the region. For a plate with density δ, the moment is M_y = ∫∫ x δ dA, where dA is the differential area element.
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Setting up the region of integration
To find the moment, the region bounded by the curve y = 36/(2x + 3), the line x = 3, and the coordinate axes must be clearly defined. This involves determining the limits for x and y, ensuring the integration covers the entire plate in the first quadrant. Proper limits are essential for accurate calculation.
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Double integration for area and moments
Double integrals allow calculation of quantities over two-dimensional regions. Here, integrating δ x dA over the plate's area sums the contributions of all small elements. The integral is often set up as ∫ from x=0 to 3 ∫ from y=0 to y=36/(2x+3) of x δ dy dx, simplifying the evaluation of the moment.
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Related Practice
Textbook Question
Evaluate the integrals in Exercises 1–14.
∫ (3 dx) / √(1 + 9x²)
Textbook Question
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ t² e^(4t) dt
Textbook Question
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 ((x dx) / √(x⁴ - 1))
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.
∫ (dx / ((x - 2)√(x² - 4x + 3)))
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Textbook Question
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ dx / √(1 - x²)