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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 6 - Applications of Integration문제 6.3.12
6장, 문제 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
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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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6m

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.
추천 영상:
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.
추천 영상:
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.
추천 영상:
04:48
Finding Volume Using Disks
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교과서 질문

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

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교과서 질문

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.

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교과서 질문

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]

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교과서 질문

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]

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교과서 질문

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.

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교과서 질문

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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