Domanda del libro di testo
Evaluate ∫ x³ √(1 - x²) dx using:
a. Integration by parts.
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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 8.2.61b
Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes and the curve y = cos(x), 0 ≤ x ≤ π/2, about
b. The line x = π/2.
Guida verificata passo dopo passo1
Identify the region to be revolved: it is bounded by the x-axis (y=0), the y-axis (x=0), and the curve \(y = \cos(x)\) for \(0 \leq x \leq \frac{\pi}{2}\), all in the first quadrant.
Since the solid is generated by revolving the region about the vertical line \(x = \frac{\pi}{2}\), use the method of cylindrical shells to find the volume.
Set up the volume integral using cylindrical shells: the radius of a shell is the horizontal distance from \(x\) to the line \(x = \frac{\pi}{2}\), which is \(r = \frac{\pi}{2} - x\), and the height of the shell is \(h = \cos(x)\).
Write the volume integral as \(V = 2\pi \int_0^{\frac{\pi}{2}} (\text{radius})(\text{height}) \, dx = 2\pi \int_0^{\frac{\pi}{2}} \left( \frac{\pi}{2} - x \right) \cos(x) \, dx\).
Evaluate the integral by applying integration techniques such as integration by parts to find the volume.
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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 a given axis. The volume can be computed using integral methods such as the disk, washer, or shell method, depending on the axis of rotation and the shape of the region.
Video consigliato:
04:48
Finding Volume Using Disks
Shell Method
The shell method calculates volume by integrating cylindrical shells formed by revolving vertical or horizontal slices around an axis. It is especially useful when the axis of rotation is vertical and not the y-axis, as in this problem where the region is revolved around x = π/2.
Video consigliato:
07:33Euler's Method
Trigonometric Functions and Their Graphs
Understanding the behavior of y = cos(x) on the interval [0, π/2] is essential. The cosine function decreases from 1 to 0 in this range, defining the boundary of the region to be revolved. This knowledge helps set up correct limits and integrand expressions for volume calculation.
Video consigliato:
Percorso guidato
6:04Introduction to Trigonometric Functions
Pratica correlata
Domanda del libro di testo
90. Consider the infinite region in the first quadrant bounded by the graphs of
y = 1 / √x, y = 0, x = 0, and x = 1.
b. Find the volume of the solid formed by revolving the region (i) about the x-axis
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Domanda del libro di testo
Finding area
Find the area of the region enclosed by the curve y = x sin(x) and the x-axis (see the accompanying figure) for:
b. π ≤ x ≤ 2π.
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Domanda del libro di testo
Finding area
Find the area of the region enclosed by the curve y = x cos(x) and the x-axis (see the accompanying figure) for:
b. 3π/2 ≤ x ≤ 5π/2.
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Domanda del libro di testo
In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 0 to 3 of 1/√(x + 1) dx
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Domanda del libro di testo
In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from -1 to 1 of (x² + 1) dx
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