Domanda del libro di testo
Find the area of the region described in the following exercises.
The region in the first quadrant bounded by y=x^2/3 and y=4
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Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 6, Problema 6.3.64
Let f(x) = {x if 0≤x≤2
2x−2 if 2<x≤5
−2x+18 if 5<x≤6.
Find the volume of the solid formed when the region bounded by the graph of f, the x-axis, and the line x=6 is revolved about the x-axis.
Guida verificata passo dopo passo1
Identify the piecewise function \( f(x) \) defined as:
\[
f(x) = \begin{cases}
x & \text{if } 0 \leq x \leq 2 \\
2x - 2 & \text{if } 2 < x \leq 5 \\
-2x + 18 & \text{if } 5 < x \leq 6
\end{cases}
\]
This function describes the height of the region above the x-axis for each interval.
Recall that the volume of the solid formed by revolving the region bounded by \( f(x) \), the x-axis, and the vertical line \( x = 6 \) about the x-axis can be found using the disk method. The volume \( V \) is given by the integral:
\[
V = \pi \int_a^b [f(x)]^2 \, dx
\]
where \( a = 0 \) and \( b = 6 \) in this problem.
Since \( f(x) \) is piecewise, split the integral into three parts corresponding to the intervals:
\[
V = \pi \left( \int_0^2 (x)^2 \, dx + \int_2^5 (2x - 2)^2 \, dx + \int_5^6 (-2x + 18)^2 \, dx \right)
\]
Set up each integral explicitly:
- For \( 0 \leq x \leq 2 \):
\[
\int_0^2 x^2 \, dx
\]
- For \( 2 < x \leq 5 \):
\[
\int_2^5 (2x - 2)^2 \, dx
\]
- For \( 5 < x \leq 6 \):
\[
\int_5^6 (-2x + 18)^2 \, dx
\]
Evaluate each integral separately and then sum the results. Finally, multiply the sum by \( \pi \) to find the total volume of the solid formed by revolving the region about the x-axis.
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Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Piecewise Functions
A piecewise function is defined by different expressions over distinct intervals of the domain. Understanding how to interpret and work with each piece separately is essential, especially when integrating or analyzing the function over its entire domain.
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Domanda del libro di testo
Assume f and g are continuous, with f(x) ≥ g(x) ≥ 0 on [a, b]. The region bounded by the graphs of f and g and the lines x=a and x=b is revolved about the y-axis. Write the integral given by the shell method that equals the volume of the resulting solid.
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Domanda del libro di testo
Leaky Bucket A 1-kg bucket resting on the ground contains 3 kg of water. How much work is required to raise the bucket vertically a distance of 10 m if water leaks out of the bucket at a constant rate of 1/5 kg/m? Assume the weight of the rope used to raise the bucket is negligible. (Hint: Use the definition of work, W = ∫a^bF(y) dy, where F is the variable force required to lift an object along a vertical line from y=a to y=b.)
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Domanda del libro di testo
Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given line.
x=2−secy,x=2,y=π/3, and y=0; about x=2
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39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.
x =2
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Domanda del libro di testo
13–20. Mass of one-dimensional objects Find the mass of the following thin bars with the given density function.
ρ(x) = {1 if 0≤x≤2 {2 if 2<x≤3
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