FINDING VOLUME OF A SOLIDDiscuss the three methods for finding volumeof a solid obtained by rotating a region about agiven axis. Consider the function f (x) =%3DV4 – x defined over the interval [0, 4] isrotated around the x-axis Discuss which of thethree methods can be used to find the volumeof the solid and set up the formula for findingthe volume.

Answered: Image /qna-images/answer/cec8942e-ccd3-4d9f-a1a1-38d0670064fe.jpg

Discuss the three methods for finding volume of a solid obtained by rotating a region about a given axis. Consider the functionf (x) = V4 – x defined over the interval [0, 4] is %3D rotated around the x-axis Discuss which of the three methods can be used to find the volume of the solid and set up the formula for finding the volume.

Discuss the three methods for finding volume of a solid obtained by rotating a region about a given axis. Consider the functionf (x) = V4 – x defined over the interval [0, 4] is %3D rotated around the x-axis Discuss which of the three methods can be used to find the volume of the solid and set up the formula for finding the volume.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section: Chapter Questions

Problem 80RE

See similar textbooks

Related questions

Q: Find the volume of the solid generated by revolving the region bounded by y = 3x, y = 0, and x= 2…

Q: Find the volume of a solid, whose base is the region bounded by the graph of |1–(In ¤)² g(z)=1 and…

Q: 1. Volume of a hemisphere Derive the formula V = (2/3)™R³ for the volume of a hemisphere of radius R…

Q: (10) Find the volume of a solid S where the base of S is the region enclosed by the parabola y = 1x²…

A: Solution: lets take the integral over the y axis. We will notice that y=1-x2 is an upside…

Q: r = b, the graph of y = f(x) and the x-axis, and ions of the solid perpendicular to the x-axis are…

A: Here it is given that, a solid is constructed so that, one side is the region between the lines x=a…

Q: 12(x2 + y²) and above the The volume of the solid under the surface z = region 22 + y 0, is a) None…

A: Given problem:- The volume of the solid under the surface z = 12(x2 + y2)and above the region x2 +…

Q: 7. Consider the region above the r-axis and below y = e-² with x > 1, shown below. 4 Consider a…

Q: Solids of revolution Determine the volume generated about x-axis the area enclosed by 4x - y² = 4, x…

A: To find the volume of solid formed when the region bounded by 4x - y² = 4, x = 2 and x-axis revolved…

Q: (a) If the region shown in the figure is rotated about the x-axis to form a solid, use the Midpoint…

A: To determine :- The volume if the region is rotated about the y-axis by the midpoint rule with n=4.…

Q: Find the volume of the solid generated by revolving about the y-axis the region bounded by the given…

A: The volume of solid of revolution between curve f(y) and curve g(y) rotated around x=0 (y-axis) on…

Q: onsider the region bounded by y = x^3, y = 8, and x = 0. Find the volume of the solid using…

A: Given: A region bounded by y=x3, y=8 and x=0. To find the volume of the solid using the cylindrical…

Q: Solids of revolution Determine the volume generated about y-axis the area enclosed by: y²- 4y + 2x =…

A: Since you have asked for a multiple question, we will solve the first question for you. If you want…

Q: Suppose that R is the finite region bounded by f(x) = 4/x and g(x) = x. Find the exact value of the…

Q: The distance from the center of the rectangle to the axis of revolution is p(x) = x, and the height…

Q: 1-8 Find the volume of the solid that results when the shaded region is revolved about the indicated…

A: Since you have posted multiple questions we solve first question for you To get the solution of…

Q: 6. Consider the region R bounded by f(x) = (9+ x²)-1/2 and the x-axis on the interval [0,4] (a) Find…

A: “Since you have asked multiple questions in a single request, we will be answering only the 1st…

Q: Find the volume of the solid generated by revolving the region bounded by y = 4x and the x-axis from…

Q: Find the volume of R bounded by y = e^-x, y = 0, x=0, x = ln 11. Show how you would find the radius,…

A: the volume of R bounded by y = e^-x, y = 0, x=0, x = ln 11.

Q: 10. The base of S is the triangular region with vertices (0,0), (1,0), and (0,1). Cross-sections…

Q: A region bounded by f(x) = 3 − x, y = 1, and x = 0 is shown below. Find the volume of the solid…

Q: Find the volume of the solid obtained by rotating the region underneath the graph of the function…

A: Given: f(x ) = x2+25 limit : [ 0 , 1 ]

Q: 4. Consider the region R in the first quadrant bounded by the graphs of y = x-1 and x = 5. a.) Find…

Q: For the region bounded by y = x, the x-axis, and x= 1, determine which of the following is greater:…

Q: Suppose that R is the finite region bounded by f(x) = Va and g(x) 4 Find the exact value of the…

Q: A circle of radius a whose center is at (0, k ) k is rotated around the x-axis. Calculate the volume…

Q: Suppose that R is the finite region bounded by f(x) Væ and f(x) 3 Find the exact value of the volume…

Q: 1-8 Find the volume of the solid that results when the shaded region is revolved about the indicated…

Q: Suppose that R is the finite region bounded by f(x) = Va and f(x) : 3 Find the exact value of the…

A: Graph the given region as follows. Observe that the limits are as follows. 0≤x≤90≤y≤3

Q: Let f and g be functions defined by f (x) = 1+x +ex-2x and g(x) = x4 – 6.5x² + 6x + 2. Let R and S…

A: Consider the two regions R and S enclosed by the graphs of the given functions as shown in the…

Q: a) Determine the intersection points. b) Draw a sketch of the graph. c) Find the volume of the solid…

Q: 9) Given the finite region f(x)=- x=1, x=2 Vx a) Find the area of the region b) Find the volume if…

Q: On the interval 0 < x < T, spin the area enclosed by f (x) and the x-axis around the y-axis. Find…

Q: (c) It is possible to compute the volume of a solid of revolution using disk/washer method if and…

A: Consider the provided question, According to you we need to solve only part (e). (e) Given, It is…

Q: 7. Consider the region above the x-axis and below y = e-² with x > 1, shown below. 4 Consider a…

Q: 1-8 Find the volume of the solid that results when the shaded region is revolved about the indicated…

A: Note: Since you have asked multiple question, we will solve the first question for you. If you want…

Q: V= , y = 3.4r' +0.8 and a-1

A: Express the graph of the region. From the graph, dy=e3.4x-3.4x2-0.8 Since the cross section…

Q: 2. Find the volume of the described solid of revolution or state that it does not exist. (a) The…

Q: 2. Given are functions g(x) =x and h(x) = 1–. The area enclosed by these two functions, the line x =…

A: It is given that gx=x4 and hx=1-1x. We have to evaluate the volume of the solid of revolution by…

Q: The graph of f (x) = 9e¬3x, thex-axis, the y-axis, and the line x = 2 form a bounded region. Find…

Q: The distance from the center of the rectangle to the axis of revolution is p(X) X, and the height of…

Q: = 0x5 on the Q7. Compute the volume of the solid obtained by revolving the graph of the function…

A: I have attached a photo copy below, please check it.

Q: The region bounded by the given curves is rotated about the specified axis. Find the volume of the…

Q: Let R be the region bounded above by the graph of y = e02, below by the x-axis, and on the left by…

Q: The region bounded by y = e*, y = 0, x = 0, and x = b (b > 0) is revolved about the y-axis. (Round…

A: We solve the given problem by using washer's method as follows :

Q: Suppose that R is the finite region of the object by f(x)=square root x and f(x)=x/2 (a)Find the…

Q: 1-8 Find the volume of the solid that results when the shaded region is revolved about the indicated…

A: Given,

Q: 3) The solid S obtained by rotating the region below the graph of y = x about the x-axis for 1 < x…

Q: (a) Show that the volume of the solid in R that lies below the surface z = f(x, y) and above the…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Find the volume of the solid generated by revolving about the y-axis the region bounded by the given…

A: First we should recognise the small volume dv and then integrate it between the appropriate limits.

Concept explainers

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.

Question

FINDING VOLUME OF A SOLID
Discuss the three methods for finding volume
of a solid obtained by rotating a region about a
given axis. Consider the function f (x) =
%3D
V4 – x defined over the interval [0, 4] is
rotated around the x-axis Discuss which of the
three methods can be used to find the volume
of the solid and set up the formula for finding
the volume.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step by step

Solved in 4 steps with 4 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Application of Integration$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Similar questions

Let f(x) = log(x2 + 1), g(x) = 10 – x2, and R be the region bounded by the graphs of f and g, as shown above.a) Find the volume of the solid generated when R is revolved about the horizontal line y = 10.b) Region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is anisosceles right triangle with a leg in R. Find the volume of the solid.c) The horizontal line y = 1 divides region R into two regions such that the ratio of the area of thelarger region to the area of the smaller region is k:1. Find the value of k.Solve all three
A tank on the wing of a jet aircraft is formed by revolving the region bounded by the graph of y = (1/ 8)x2 √(2 − x) and the x-axis (0 ≤ x ≤ 2) about the x-axis, where x and y are measured in meters. Use a graphing utility to graph the function. Find the volume of the tank analytically.
Find the area of the region. f(x) = 16 − x2 The x y-coordinate plane is given. There is 1 curve and a shaded region on the graph. The curve enters the window in the third quadrant, goes up and right becoming less steep, crosses the x-axis at x = −4, changes direction at the point (0, 16), goes down and right becoming more steep, crosses the x-axis at x = 4, and exits the window in the fourth quadrant. The region below the curve, above the x-axis, and between −4 and 4 on the x-axis is shaded.
SOLID OF REVOLUTION Find the volume of the solid of revolution of the section bounded by the line y=4x and the parabola y=x2 rotated with the x-axis as the access of the revolution. Solve using the disk and washer method. Solve again using the shell method.
R is the region bounded by y= sqrt x , and y= 6-x and the vertical y-axis. Q is the region bounded by y= sqrt x, and y=6-x, and the horizontal x- axis. (They Intersect at (4,2) ) Write an expression that gives the volume of the solid formed by rotating R around the horizontal line y=-2
[Areas of Integration]Given the following graphs, find the volume of the solid of revolution of the shaded region.Rotate about the y-axis, using ring method and cylindrical shell method
Find the area of the region. f(x) = x-3/x The x y-coordinate plane is given. There is 1 curve and a shaded region on the graph. The curve starts at x = 3 on the x-axis, goes up and right becoming less steep, and ends at the approximate point (5, 0.40). The region below the curve, above the x-axis, and between 3 and 5 on the x-axis is shaded.
function and the region bounded by the x-axis around the x-axisbeing rotated.By plotting the graph of the object formed,Find its volume using the shell method.
[Areas of Integration]Given the following graphs, find the volume of the solid of revolution of the shaded region.Rotate about the x-axis.
Find the volume of the solid obtained by rotating the region enclosed by the graphs about the axis. f(x)=16/x^2, g(x)=17−x2, x≥0, about x=−1 (Use symbolic notation and fractions where needed.)
Volume of a solid obtained by revolving about the y-axis the region bounded above by y=1/sqrroot(x) on the left by the line x=1/4 and below by y=1.
volume of the solid generated when the region bounded by y = 9 − x2 and y = 2x + 6 is revolved about the x-axis.