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Geometry and Topology

Shell method of finding volume of revolving solid

Revolving the solid about the y-axis, use the shell method to find the volume of the solid bounded by: the line x=√2/2 on the right y=1/√(1-x²) above. The shaded area in the drawing is bounded by the y-axis to the left, and the x-axis on the bottom.

Shell method to find volume about x-axis

Use the shell method formula to find the volume of the solid generated by revolving the shaded region bounded by the curves and lines below about the x-axis: V=∫2π(shell radius)(shell height)dy = ∫2πx f(y)dy a≤y≤b Shaded region:: x=y² x= -y y=2

Volume of a solid of revolution

Find the volume of the solid generated by revolving the region described about the Y axis: Between (0,0) and (0,2), the triangular region between those points on the y-axis and the straight line x=3y/2 using the formula V=∫π[R(y)]²dy

The surface area paint needed

--- The surface area A of a steel cylinder is given by the formula A= 2.pi.r²+2.pi. r h Where r is the radius and h the height a) find the required radius if A = 12500mm² , and h = 150mm b) Determine the number of 2.5 liter tins of paint that are needed to coat 500 cylinders with a thickne

Kilometers and starting point

Can you tell me what the answer would be if I traveled north for 35 kilometers then traveled east 65 kilometers. How far am I from my starting point? using: 35^2+65^2?

Relation in radius and height of a cylinder of given volume

Suppose the volume of a cylinder (think about the volume of a can) is given by V = πr2h where r is the radius of the cylinder and h is the height of the cylinder. Now suppose the volume of the can is 100 cubic centimeters. How do would you write h as a function of r? What is the measurement of the height if the radius o

Please explain in steps how to complete the following problem

Find sin2x, cos2x, and tan 2x under the given conditions. 23. sinx=5/13 (0<x<pie/2) ans. sin 2x = 120/169, cos 2x = 119/169, tan 2x = 120/119 Please explain in detail step by step how to come up with this ans. 25. cos x=-3/5 (pie<x< 3pie/2) ans. sine 2x=24/25, cos2x=-7/25, tan2x=-24/7

Frieze patterns

Q 1. A frieze pattern has one and only one direction of translation. The translation isometry is denoted by T. As we have noticed in lectures the symmetry group of the fundamental pattern consists of all powers of T and is thus isomorphic to which group? Q 2. Now use Lemma F from your lecture notes to prove the following r

Diagonals, Squares Circles and Endpoints

6. If a square's diagonal has endpoints (3,4) and (7,8) , find the endpoints of the other diagonal. (please illlustrate) What is the length of the diagonal? (Please give formula) What is the perimeter of the square? Write the equation of the circle that has the endpoint (3,4) as its center and goes throught the two c

Can you lease check my work

I have attached a file regarding contrapositives and inverse and converse statements --- Statement: If two lines are parallel then they do not intersect. True Converse: If 2 lines do not intersect then they are parallel False Inverse: If 2 lines are not parallel then they intersect

Space of functions is sequentially compact

Let C_0 be the space of functions f:R --> R such that lim f(x) = 0 as x goes to infinity and negative infinity C_0 becomes a metric space with sup-norm ||f|| = sup { |f(x)| : x in R } Prove that if A is a family of functions in C_0 such that A is uniformly bounded and equicontinuous, then every sequence of functions

Trivial Topology, Continuity and Connectedness

Let X and Y be topological spaces, where the only open sets of Y are the empty set and Y itself, i.e., Y has the trivial topology. ? Show that any map X --> Y is continuous ? Show that Y is path connected and simply connected. ---

Time And Distances

The movement of two submarines are being followed by a tracking system, and the positions of the submarines are modelled by points. The position of Sub A at time t is (2t+2, 2t+1) and the position of Sub B at time t is (4-t, t+5) (distance in Kilometers) 1.How would I go about eliminating t from each pair of coordinates, and


(See attached file for full problem description with proper symbols) --- Let and a map, given by . Let ~ be the equivalence relation on Xx[0,1] defined by and all other points are equivalent only to themselves. Show that Xx[0,1]/~ is homeomorphic to the Moebius strip. ---

Continuous and identification maps

(See attached file for full problem description with proper symbols and equation) --- Let be a surjective continuous map between topological spaces. Show that: a) If f is an identification mp, then for any pace Z and any map the composition is continuous if and only if g is continuous. b) If, for any space Z and any

Identification map

(See attached file for full problem description with proper symbols and equations) --- ? Let be the subspace of of all positive real numbers. Show that the map defined by is an identification map. ---

Break Even Point and Capital Budgeting

2. (Payback period, net present value, profitability index, and internal rate of return calculations) You are considering a project with an initial cash outlay of $160,000 and expected free cash flows of $40,000 at the end of each year for 6 years. The required rate of return for this project is 10 percent. a.) What is the pr


Compact spaces and path connectedness (see attachment). --- ? Show that if is a homeomorphism between topological spaces, then X is path connected if and only if Y is path connected. Using open cover definition: 1) is a compact subset? 2) Is a compact subset? ---

Connectedness, Continuity, Image, Antipodal Point and Borsuk-Ulam Theorem

Show that, if X is a connected topological space and is continuous, then the image f(X) is an n interval. Show that, if is a continuous map, then if given a,b,c in with a < b and c between f(a) and f(b), there exists at least one with a and f(x)=c Let be a continuous map. Show that there exists a point in the ci

Compact Set, Convergent Sequences and Subsequences and Accumulation Points

Prove that a set A, a subset of the real numbers, is compact if and only if every sequence {an} where an is in A for all n, has a convergent subsequence converging to a point in A. For the forward direction, I know that a compact set is closed and bounded, thus every sequence in A is bounded, and so has a convergent subsequen


Exactly how many minutes is it before eight o'clock, if 40 minutes ago, it was three times as many minutes past four o'clock?

Use double inegrals to find the volume of a region.

Find the volume of the following region in space: The first octant region bounded by the coordinate planes and the surfaces y=1-x^2, z=1-x^2. This question is #12 (section 9.3) in Advanced engineering mathmatics (8th ed.) by Kreyszig. This section deals with the evaluation of double integrals.

Use the cylindrical shell method to find volume of solid of revolution.

Show step by step work using cylindrical shell method to find the volume of the solid formed by revolving the given region about the y-axis. 22) the region bounded by the curve y=SQRT(x), the y-axis, and the line y=1. 24) the region bounded by the parabolas y=x^2, y=1-x^2, and y axis for x&#8805;0. 26) the region ins