Domanda del libro di testo
2. When applying the formula for integration by parts, how do you choose the u and dv? How can you apply integration by parts to an integral of the form ∫ f(x) dx?
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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
Capitolo 8, Problema 8.AAE.13
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.
Guida verificata passo dopo passo1
Identify the region bounded by the curve \(y = 3x\sqrt{1 - x}\), the x-axis, and the first quadrant. Since the region is in the first quadrant, \(x\) ranges from 0 to 1 because \(\sqrt{1 - x}\) is real and non-negative only for \(0 \leq x \leq 1\).
Since the solid is generated by revolving the region about the y-axis, consider using the method of cylindrical shells. The formula for the volume using cylindrical shells is:
\(V = \int_a^b 2\pi x \cdot f(x) \, dx\)
where \(f(x)\) is the height of the shell and \(x\) is the radius.
Substitute the function \(f(x) = 3x\sqrt{1 - x}\) into the volume integral:
\(V = \int_0^1 2\pi x \cdot 3x\sqrt{1 - x} \, dx = \int_0^1 6\pi x^2 \sqrt{1 - x} \, dx\).
Simplify the integral expression and prepare to evaluate it:
\(V = 6\pi \int_0^1 x^2 (1 - x)^{1/2} \, dx\).
To solve the integral, consider using a substitution such as \(u = 1 - x\), which will simplify the integral into a form involving powers of \(u\). Then rewrite \(x\) in terms of \(u\) and change the limits accordingly before integrating.
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Volume of Solids of Revolution
This concept involves finding the volume of a 3D solid formed by rotating a 2D region around an axis. Common methods include the disk/washer method and the shell method, which use integration to sum infinitesimal volumes. Choosing the appropriate method depends on the axis of rotation and the shape of the region.
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04:48
Finding Volume Using Disks
Shell Method
The shell method calculates volume by integrating cylindrical shells formed by revolving vertical slices around an axis. Each shell's volume is approximated by 2π(radius)(height)(thickness). This method is especially useful when revolving around the y-axis and the function is given in terms of x.
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07:33Euler's Method
Integration of Functions Involving Radicals
Integrating functions with radicals, such as y = 3x√(1 − x), often requires substitution to simplify the integral. Recognizing suitable substitutions helps transform the integral into a standard form, making it easier to evaluate and find exact volumes.
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07:01Integrals Involving Natural Logs: Substitution
Pratica correlata
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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.
∫ dx / (1 + sin x + cos x)
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Domanda del libro di testo
Finding volume
The infinite region bounded by the coordinate axes and the curve y = −ln x in the first quadrant is revolved about the x-axis to generate a solid. Find the volume of the solid.
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Domanda del libro di testo
7. What is the goal of the method of partial fractions?
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Domanda del libro di testo
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.
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Domanda del libro di testo
Use the substitution z = tan(θ/2) to evaluate the integrals in Exercises 41 and 42.
∫ csc θ dθ
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