1. Find the volume of an ellipsoid obtained by rotating the region confined between the y-axis and the graph of ellipse y² 1. a? 62 (a) Sketch carefully the graph of an ellipse. Feel free to google it, but you should learn how to provide a plot of an ellipse given arbitrary positive constants a and b. (b) Identify the region that you will rotate around the y-axis and show it on your sketch. (c) Use the method of disks to find the volume (set up the integral and evaluate it providing all the details). (d) Use the method of cylindrical shells to find the volume (set up the integral and evaluate it providing all the details). (e) Did you get the same answers in the two previous points? (f) Make a correct guess without any calculations about the volume of an ellipsoid obtained by rotating around the x-axis the region between the x-axis and the graph of the same ellipse.

1. Find the volume of an ellipsoid obtained by rotating the region confined between the y-axis and the graph of ellipse y² 1. a? 62 (a) Sketch carefully the graph of an ellipse. Feel free to google it, but you should learn how to provide a plot of an ellipse given arbitrary positive constants a and b. (b) Identify the region that you will rotate around the y-axis and show it on your sketch. (c) Use the method of disks to find the volume (set up the integral and evaluate it providing all the details). (d) Use the method of cylindrical shells to find the volume (set up the integral and evaluate it providing all the details). (e) Did you get the same answers in the two previous points? (f) Make a correct guess without any calculations about the volume of an ellipsoid obtained by rotating around the x-axis the region between the x-axis and the graph of the same ellipse.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section11.2: Ellipses

Problem 24E

See similar textbooks

Related questions

Q: An ellipse + 1 is revolved around the y-axis. The volume generated by the solid of revolution if a =…

A: Let us first calculate the y value on the ellipse at x = t. From the equation of the ellipse, at x…

Q: 4.x² + y = 4

Q: 49. Alatus rectum of a conic section is a chord through a focus parallel to the directrix. Find the…

Q: Use Green's theorem to find the area between ellipse 16 = 1 and circle x2 + y2 = 25.

Q: 2. Sketch the graphs of y = x² - 6x + 12, x = 5, x-axis and y-axis. Shade the region bounded by the…

A: The given equation of the curve is: and

Q: s + 2 2. 52 6s + 8 -

Q: What is the area of the ellipse x² 16 + y² 9 = 1?

Q: 5. Figure below shows triangle AOC inscribed in the region cut from the parabola y=x by the line…

Q: Find the volume of the solid formed if we rotate the ellipse+ = 1 about the line 4x + 3y = 20.

A: Calculation of Volume.

Q: A football is in the shape of a solid obtained by rotating an ellipse a² b² inflated NFL football…

A: Equation of ellipse x2a2+y2b2=1 Length 11.125 inches and 28.25 inches in center circumference.…

Q: Suppose we want to form a conic container by rolling up a circular sector of radius 10. We want to…

A: Given radius of circular sector =10=R Volume of a cone=13πr2h . . . . (1) Constraint is…

Q: If the ellipse is rotated about the x-axis, an ellipsoid is formed. In the ellipsoid below, segment…

A: If the ellipse is rotated about the x-axis, an ellipsoid is formed. In the ellipsoid below, segment…

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

1. Find the volume of an ellipsoid obtained by rotating the region confined between the y-axis and
the graph of ellipse
x2
= 1.
62
a2
(a) Sketch carefully the graph of an ellipse. Feel free to google it, but you should learn how to
provide a plot of an ellipse given arbitrary positive constants a and b.
(b) Identify the region that you will rotate around the y-axis and show it on your sketch.
(c) Use the method of disks to find the volume (set up the integral and evaluate it providing
all the details)..
(d) Use the method of cylindrical shells to find the volume (set up the integral and evaluate it
providing all the details).
(e) Did you get the same answers in the two previous points?
(f) Make a correct guess without any calculations about the volume of an ellipsoid obtained by
rotating around the x-axis the region between the x-axis and the graph of the same ellipse.

Expert Solution

This question has been solved!

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

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step 5

Step 6

Step 7

Trending now

This is a popular solution!

Step by step

Solved in 7 steps with 5 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, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

4) Use integration to find the area of the ellipse 4x2 + y² = 9.
. A pool, when viewed from above, is in the shape of an ellipse whose equation is given by (x ^ 2)/144 + (y ^ 2)/49 = 1 A) solve the given equation for y B) Use integration techniques from chapter 5 and/or 7 to compute the area of the ellipse. Hint: Compute the area of the part of the ellipse in quadrant I. Then, multiply your answer by 4 to get the area of the entire ellipse. The units for the area in this problem are in f * t ^ 2 C) Suppose the pool is 10 feet deep and has the shape of an ellipse throughout (an elliptical cylinder). Further, suppose the pool currently has 9 feet of water. Find the work done to pump all of the water out of the pool To do this, first set this up as a Riemann sum, finding the work W_{i} required to pump the slice of water, of width Delta*x out of the pool. Then, set up the corresponding integral and integrate to find the work done to pump all of the water out of the pool. Use the fact that water weights 62.5 lbs / f * t ^ 3
5) Using transformation and integration, find the area of the region that is enclosed by the ellipse 9x² + 16Y² = 36 is
x2 y2 = 1 and circle x2 + y2 = 25. Use Green's theorem to find the area between ellipse. 16
49. Alatus rectum of a conic section is a chord through a focus parallel to the directrix. Find the area bounded by the parabola y = x²/(4c) and its latus rectum. Axis F = (0, c) Vertex X- Directrix D y= -c Parabola with focus (0, c) and directrix y = -c.
Determine the area bounded by the ellipse satisfying:(101x+ 10y)2+ (20x+ 2y)2= 1.
Find the point (a, B) on the ellipse 4x² + 3y² = 12, in the first quadrant, so that the area enclosed by the lines y=x, y=B₁x=a₂ and the x-axis is maximum.
What is the area of the ellipse x² 16 + y² 9 = 1?
The curve y = sqrt(x2 + 1), 0<=x<=sqrt(2), which is part of the upper branch of the hyperbola y2 - x2 = 1, is re-volved about the x-axis to generate a surface. Find the area of the surface.
find the surfase area of ellipsoid obtained by yevoiving the upper-half of the ellipse x2/a2+y2/b2=1.about x-axis.given that a2-b2=1?
A rectangle with sides parallel to the x axis and y axis, and symmetric with respect to the y axis, is to be inscribed in the ellipse (x^2)/4 +y^2 = 1. Find the maximum possible area A of the rectangle, and support your answer.