Solve Interval Notation

The interval notation denotes the set of real datas with the property that any data that deception among in the set is also has in the set. For instance, the group of all the datas x satisfying 0 ≤ x ≤1 this is an range which has 0 and 1, as well as all numerical values among 0 and 1.

We can represent the inequality result by solving the interval notation.
Solving Interval Notation:

We can express the inequality outcomes by solving the interval notation

The symbols used in solving the interval notation are,

( - is the “not included” symbol or “open” symbol

[ - is the “included” symbol or “closed” symbol

Solving Open Interval:

(p, q)  is written as p ≤ x ≤ q  here the termination points are not included.

open interval

Solving Closed Interval:

[p, q]  is written as p < x < q  here the termination points are included.

closed interval

Solving Half-Open Interval:

(p, q]  is written p < x
half-open interval

Solving Half-Closed Interval:

[p, q) is taken as p < x
half-closed interval

Solving Non-ending Interval:

( p , ∞) is taken as x > p where p is not incorporated and infinity is always defined as being "open".

non-ending interval

Solving Non-ending Interval:

( -∞ , q ] is interpreted as x < q here q is incorporated and again, infinity is always defined as being "open".

non-ending interval1

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More about Solving Interval Notation:

If we accomplish the favored set of outcomes we can use the concoction of interval notations. For example define the interval of all values except 9.

As an inequality  x<9 or="or" x="x">9

In an interval notation ( -∞ ,9) U (9,∞).

Scalar Line Integrals

Scalar line integral is a definite integral it will be taken over a surface and integrated, so the scalar line integral is defined as the sum of all points in the surface.  Let x: [a, b] gives R3 be a path of class C and f: X subeR3 gives R be a continuous function contains the image of x.  The scalar line integral of f along x is

int_a^bf(x(t)) || x'(t) dt

Notation is usually written as int_x f ds.

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General form of scalar line integral:

The scalar function of line integral  is

if F = F1 i + F2 j + F3 k

r = x i + y j + z k

So int_cF. dr = int_c (f_(1)dx + f_(2) dy + f_(3) dz)

Scalar line integral and parametrizations

If y is a reparametrization of x. Then

If y is orientation-preserving, then int_y F.ds =  int_x F.ds

If y is orientation-preserving, then  int_y F.ds  = -int_x F.ds


When the path is parametrized by length of arc, the natural analog of the integral has done in 1 dimension. The integral of a scalar function f along a curve r(s) is simply int f (r(s)) ds

Applications:

F is a force acting upon a the particular particle so the particle moves along a curve C in  sample space and r be the position vector of the  given particle at a point on C. Then work done by the given particle at C is F.dr and the total work is done by F along a curve C is given by the line integral  int_c F. dr



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Scalar Line Integrals don’t depend on parameterizations.
Scalar Line Integrals-example Problems

Evaluate I = int_e f(x, y, z) ds where f(x, y, z) = = x2 – y2 – 1 + z and e is the helix parametrized by c(t) = (cos t, sin t, t)  [o <= t <=pi]. 3

Solution:

I = int_alpha^beta f(c(t)) || c'(t) || dt

See that f (c (t)) = cos2t + sin2t - 1

Also  c' (t) = (-sin t, cos t, 1) =    sqrt(-sin t^(2) + cos^(2) t + 1^(2))

See that f(c (t)) = cos2t + sin2t - 1

I = int_0^(3pi)sqrt(2) t dt

= sqrt(2) [(1)/(2)  t^(2) ][]_0^(3pi)

= (sqrt(2))/(2) 9pi^(2).

The correct answer for scalar line integral is = (sqrt(2))/(2) 9pi^(2).

Practice Problem:

Evaluate int sin 6x cos 3x dx


Answer: -1/2 [(cos 9x / 9) + (cos 3x / 3)] + c.