Textbook Question
{Use of Tech} Decreasing velocity A projectile is fired upward, and its velocity (in m/s) is given by v(t) = 200 / √t+1, for t≥0.
a. Graph the velocity function, for t≥0.
Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 6, Problem 6.R.66c
Surface area and volume Let f(x) = 1/3 x³ and let R be the region bounded by the graph of f and the x-axis on the interval [0, 2].
c. Find the volume of the solid generated when R is revolved about the x-axis.
Verified step by step guidance1
Identify the region R bounded by the curve \(f(x) = \frac{1}{3}x^{3}\) and the x-axis on the interval \([0, 2]\). Since \(f(x) \geq 0\) on this interval, the region lies above the x-axis.
Recall the formula for the volume of a solid of revolution generated by revolving a region around the x-axis: \(V = \pi \int_{a}^{b} [f(x)]^{2} \, dx\).
Substitute the given function and interval into the formula: \(V = \pi \int_{0}^{2} \left( \frac{1}{3}x^{3} \right)^{2} \, dx\).
Simplify the integrand: \(\left( \frac{1}{3}x^{3} \right)^{2} = \frac{1}{9} x^{6}\), so the integral becomes \(V = \pi \int_{0}^{2} \frac{1}{9} x^{6} \, dx\).
Set up the integral for evaluation: \(V = \frac{\pi}{9} \int_{0}^{2} x^{6} \, dx\). The next step would be to compute this integral and then multiply by \(\frac{\pi}{9}\) to find the volume.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Volume of Solids of Revolution
This concept involves finding the volume of a 3D solid formed by rotating a 2D region around an axis. The volume can be computed using integral calculus, often applying the disk or washer method, which sums up infinitesimal circular cross-sections perpendicular to the axis of rotation.
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Finding Volume Using Disks
Disk Method
The disk method calculates volume by slicing the solid into thin disks perpendicular to the axis of rotation. Each disk's volume is approximated by π times the square of the radius (distance from the axis to the function) times the thickness (dx). Integrating these volumes over the interval gives the total volume.
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Definite Integration
Definite integration is used to sum continuous quantities over an interval. In this problem, it allows the calculation of the total volume by integrating the area of cross-sectional disks from x = 0 to x = 2, providing an exact value for the volume of the solid.
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Related Practice
Textbook Question
43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.
The region bounded by the curves y = sec x and y=2, for 0 ≤ x ≤ π/3, is revolved about the x-axis. What is the volume of the solid that is generated?
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Textbook Question
43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.
What is the volume of the solid whose base is the region in the first quadrant bounded by y = √x,y = 2-x, and the x-axis, and whose cross sections perpendicular to the base and parallel to the y-axis are semicircles?
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Textbook Question
Surface area of a cone Find the surface area of a cone (excluding the base) with radius 4 and height 8 using integration and a surface area integral.
Textbook Question
27–33. Multiple regions The regions R₁,R₂, and R₃ (see figure) are formed by the graphs of y = 2√x,y = 3−x,and x=3.
Find the volume of the solid obtained by revolving region R₂ about the y-axis.
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Textbook Question
14–25. {Use of Tech} Areas of regions Determine the area of the given region.
The region bounded by y = ln x,y = 1, and x = 1
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