Skip to main content
Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 6 - Applications of IntegrationProblema 6.3.12
Capitolo 6, Problema 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
fig

Guida verificata passo dopo passo
1
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 \(1\): \(V = \int_{-1}^{1} (2 - 2x^2)^2 \, dx\).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
6m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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.
Video consigliato:
05:06
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.
Video consigliato:
05:38
Introduction 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.
Video consigliato:
04:48
Finding Volume Using Disks
Pratica correlata
Domanda del libro di testo

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

69
views
Domanda del libro di testo

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.

68
views
Domanda del libro di testo

9–20. Arc length calculations Find the arc length of the following curves on the given interval.

y = x^3/2 / 3 − x^1/2 on [4, 16]

90
views
Domanda del libro di testo

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]

82
views
Domanda del libro di testo

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.

35
views
Domanda del libro di testo

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

53
views