Textbook Question
Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.
y=x^2,y=2−x, and y=0; about the y-axis
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 6.3.12
Use the general slicing method to find the volume of the following solids.
The solid whose base is the region bounded by the curves y=x^2 and y=2−x^2, and whose cross sections through the solid perpendicular to the x-axis are squares
Verified step by step guidance1
Identify the region that forms the base of the solid. The base is bounded by the curves \(y = x^2\) and \(y = 2 - x^2\). To find the interval for \(x\), set the curves equal to find the points of intersection: \(x^2 = 2 - x^2\).
Solve the equation \(x^2 = 2 - x^2\) to find the limits of integration. This simplifies to \(2x^2 = 2\), so \(x^2 = 1\), giving \(x = -1\) and \(x = 1\) as the bounds for the base region.
Determine the side length of the square cross sections perpendicular to the \(x\)-axis. The side length is the vertical distance between the curves, which is \(s = (2 - x^2) - (x^2) = 2 - 2x^2\).
Express the area of each square cross section as a function of \(x\). Since the cross sections are squares, the area is \(A(x) = s^2 = (2 - 2x^2)^2\).
Set up the integral for the volume using the slicing method. The volume is the integral of the cross-sectional area along the \(x\)-axis from \(-1\) to \(1\): \(V = \int_{-1}^{1} (2 - 2x^2)^2 \, dx\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Region Bounded by Curves
The base of the solid is the region bounded by the curves y = x² and y = 2 - x². Finding the intersection points of these curves determines the limits of integration. This region defines the domain over which the cross-sectional areas are calculated.
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Finding Area When Bounds Are Not Given
Cross Sections Perpendicular to the x-axis
Cross sections perpendicular to the x-axis are slices of the solid at a fixed x-value. Each cross section is a square whose side length is the vertical distance between the curves y = 2 - x² and y = x². The area of each cross section is the square of this side length.
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05:38Introduction to Cross Sections
Volume by Slicing Method
The volume of the solid is found by integrating the area of cross sections along the x-axis. Using the slicing method, the volume is the integral of the square of the side length of the cross section from the left to right intersection points of the curves.
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04:48Finding Volume Using Disks
Related Practice
Textbook Question
29–36. Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2).
a(t) = −32; v(0)=50; s(0)=0
Textbook Question
Express the area of the shaded region in Exercise 5 as the sum of two integrals with respect to y. Do not evaluate the integrals.
Textbook Question
9–20. Arc length calculations Find the arc length of the following curves on the given interval.
y = 3 ln x− x²/24 on [1, 6]
Textbook Question
Suppose the region bounded by the curve y=f(x) from x=0 to x=4 (see figure) is revolved about the x-axis to form a solid of revolution. Use left, right, and midpoint Riemann sums, with n=4 subintervals of equal length, to estimate the volume of the solid of revolution.
Textbook Question
35–38. Shell and washer methods Let R be the region bounded by the following curves. Use both the shell method and the washer method to find the volume of the solid generated when R is revolved about the indicated axis.
y = 8,y = 2x+2,x = 0, and x=2; about the y-axis
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