Triple integral solver

We know that when solving double integrals, we divide the two dimensional region into very small rectangles. Then we multiply the area of the rectangles with the value of the function at that point. Then we sum up these areas and then apply the following limit: lim (size of rectangle -> 0). Doing all that gives us the double integral of the said function.

Let us now try extending this concept to three dimensions. Just like in double integrals we had some region in a plane (say the xy plane), in triple integral we will consider a solid in space (i.e. xyz space). Just like in double integrals we had split the region into rectangles, in solving triple integrals we break down the solid into numerous rectangular solids (or cuboids). Extending further on same lines, just like how in double integrals we multiplied the value of the function by the area of the respective rectangle, in a triple integral example, we would multiply the value of the function at each of the point by the volume of the rectangular solid at that point. Instead of the limit of size of rectangle tending to zero, in case of triple integrals we have the limit as the volume of rectangle tending to zero.

With that we come to the definition of a triple integral. Which is like this: Triple integral is defined as the limit of the sum of product of volume of rectangular solids with the value of the function.
We call the double integral as an equivalent to double iterated integral. In the same way we can understand the triple integral as a triple iterated integral.

Symbolically the definition of a triple integral can be stated as follows:
Consider a function f(x,y,z). It is of three variables. It is continuous over any solid S. Then the triple integral of this function over the solid S can be symbolically stated as:

where the sum is taken over all the rectangular solids that are contained in the solid S. The limit is for the side length of the rectangles.
The above definition of triple integrals is useful only when we are given a set of data in the form of a table of values of volume and value of function. When a function is defined symbolically, then we use the Fubini’s theorem to solve triple integral examples.

First and Second Fundamental theorem of calculus

First Fundamental Theorem of Calculus: If f is continuous on [a, b], then F(x) = integral a to x f(t) dt has a derivative at every point of [a, b] and dF/dx = d/dx integral a to x f(t) dt = f(x), a is less than and equal to x is less than equal to b. Let us understand Proof of Fundamental Theorem of Calculus: We prove the fundamental theorem of calculus by applying the definition of derivative directly to the function F(x).

This means writing out the difference quotient F(x + h) – F(x) /h and showing that its limits as h approaches to 0 is the number f(x). When we replace F(x + h) and F(x) by their definite integrals, the numerator in above equation becomes F(x + h) – F(x) = integral a to x+h f(t) dt – integral a to x f(t) dt. The additive rule for integrals simplifies the right hand side to Integral x to x+h f(t) dt So that the above equation becomes  F(x + h) – F(x) /h = 1/h [F(x + h) – F(x)] = 1/h integral x to x+h f(t) dt. According to the mean value theorem for definite integrals, the value of the last expression in the above equation is one of the values taken on by f in the interval joining x and x + h. That is for some number c in this interval, 1/h integral x to x + h f(t) dt = f(c).

We can therefore find out what happens to (1/h) times the integral as h approaches to 0 by watching what happens to f(c) as h approaches to 0. As h approaches to 0, the endpoint x + h approaches x, pushing c ahead of it like a bead on a wire. So c approaches x, and since f is continuous at x, f(c) approaches f(x) .Lim h approaches 0 f(c) = f(x).Going back to the beginning, then we have dF/dx = lim h approaches to 0 [F(x+h) – F(x)] /h= lim h approaches 0 1/h integral x to x+h f(t) dt = lim h approaches 0 f(c)= f(x). This concludes the proof.

Let us more understand this through Fundamental Theorem of Calculus problems: let us take few fundamental theorem of calculus examples .suppose we have Find dy/dx if y = integral 1 to x^2 cos t dt.

Now to understand this solution suppose let us notice that the upper limit of integration is not x but x^2. To find dy/dx we must therefore treat y as the composite of y = integral 1 to u cos t dt and u = x^2 and apply the chain rule: dy/dx = (dy/du).(du/dx) = d/du integral 1 to u cos t dt . du/dx = cos u . du/dx= cos x^2 . 2x = 2x cos x^2.

Second Fundamental Theorem of Calculus: If f is continuous at every point of [a, b] and F is any anti-derivative of f on [a, b], then Integral a to b f(X) dx = F(b) – F(a). Let us understand this by second Fundamental Theorem of Calculus Examples suppose we have Evaluate integral 0 to pi cos x dx. Now let us solve this integral 0 to pi cos x dx = sin x 0 to pi = sin pi – sin 0 = 0 – 0 = 0.

Derivative of Logarithmic Functions

Derivative of Log Function
An exponential function is given by y=b^x, an inverse of an exponential function is given by x=b^y. The logarithmic function with base b is the function given by y=log x with base b where b>1 and the function is defined for all x greater than zero. The system of natural logarithm has a number e as its base. We know that the natural logarithms functions and natural exponential functions are inverses. If f(x) and g(x) are inverses of each other then the derivatives of inverses is given by g’(x)=1/f’[g(x)].So, the Derivative of Log Function can be found using the definition of derivatives of inverses as follows, given
f(x)=e^x and g(x) = ln x then, g’(x) = 1/f’[g(x)]
                                     =1/e^[g(x)]
                     = 1/e^ln(x)                 [g(x)=ln x]
                     = 1/x
So, the Derivative of Log Function is, d/dx[ln x]=1/x where x is greater than zero

Derivative of Log Base a of X 
We can find the Derivative of the general logarithmic function by the method of change of base formula.
 Log base a of x  can be written as ln x/ln a.
 Differentiating on both sides gives,
d/dx [log base a of x]= d/dx [ln x/ln a]
                                        =1/ln a . d/dx [ln x]
                                = 1/x ln a
Derivative of Log Base a of X is given by, d/dx[log base a of x] =1/[x ln a]
For example, find the derivative of log base 2 of x .
By the base change formula, we get, log base 2 of x = ln x/ln 2
Taking the derivative on both sides, d/dx [log base 2 of x] = d/dx [ln x/ln 2]
Here, ln 2 is a constant. On simplifying, we get, 1/ln 2 . d/dx [ln x]  
We know that derivative of ln x is 1/x. Substituting this value, we get
                        d/dx[log base 2 of x] = 1/[x ln 2]

Derivative of Log Base 10
Logarithms to base 10 are called the common logarithms. Common Logarithm is written as log x which means log base 10 of x.  Given y=ln x its inverse is given by e^y = x. Let us find the Derivative of Log Base 10 by taking derivatives on both sides with respect to x, d/dx [e^y] = d/dx[x] which gives  e^y dy/dx = 1. We have e^y= x, substituting this value, we get x dy/dx=1. We need dy/dx, so we shall divide on both sides with x which gives, dy/dx = 1/x. Derivative of Log Base 10 is given by, d/dx [log base 10 of x] = 1/x

Inverse Trigonometric Functions

Consider a function f(x)=(x+2), we can write this as y=x+2. Solving for x, we get, x = y-2. Interchanging the x and y terms gives a new function y=x-2, consider this as g(x). This new function g(x) is the inverse function of f(x). Inverse function is denoted as f^-1

Inverse Trigonometric Functions
Let us assume a given radian angle pi/2, we can evaluate a trigonometric function sin(pi/2)  equals 1. Inversely, if we are given 1 as the value of a sine function, then we can arrive to the radian angle y. Equating,  sin(pi/2) = 1, the radian angle for which the value of sine is 1 is pi/2. We can now write the inverse trigonometric function of sine as arc[sin(1)]= pi/2. In general, given the value x of the trigonometric function we can write the inverse trigonometric function as arc[sin(x)], arc[cos(x)],arc[tan(x)], arc[csc(x)], arc[sec(x)], arc[cot(x)] to name the radian angle that has the value x in each case.

Using the derivatives of inverse trigonometric identities we can obtain inverse trig functions integrals:

Integral[du/sqrt(a^2-u^2)] = sin^-1[u/a] +c
Integral[du/(a^2+u^2)=1/a[tan^-1(u/a)]+c
Integral[du/u[sqrt(u^2-a^2)]=1/a[sec^-1mod(u/a)] +c;
where u is a function of x, which is written as u=f(x)

How to Solve Inverse Trig Functions
Steps involved to solve inverse trig function, arc tan(-1):
Step1:  let y=arc tan(-1)
Step2:  tan (y) is -1 when pi/2<=y<=pi/2 [finding the interval]
Step3:  determine the radian angle of tan that gives the value of 1, which is pi/4
Step4:  determine the radian angle for a negative value, we know that tan(-x)=- tan(x)
Step5:  we get, tan y = -1
Step6:  the required radian angle is y = pi/4

Inverse Trig Functions Problems
Evaluate arc cos(-1/2)
Let y = arc cos(-1/2), we need to find the radian angle y
cos(y) = -1/2 , we know that pi/3 radian angle of cos gives the value 1/2. But here we have -1/2
We know that, cos(pi-x) gives us –cosx.
So, cos(pi-pi/3) = -1/2
We get, y=pi – pi/3 = 2pi/3 is the required radian angle

How to Graph Inverse Trig Functions
To graph inverse trig functions, first we need to understand the domain and range of the inverse trig functions which are:
Function Domain Range
y=arc sin(x)  -1<= x<= 1    -(pi/2)<=y<=(pi/2)
y=arcos(x)   -1<= x<=1   0<=y<=pi
y=arctan(x)   -infinity
y=arccsc(x)  x<=-1or x>=1  -pi/2<=y<=pi/2, y not equals zero
y=arcsec(x)   x<=-1 or x>=1  0<=y<=pi, y not equals pi/2
y=arccot(x)      -infinity  0

Plug in different values of x (within the range) to arrive to the respective y values of the given inverse trig function. Plot the coordinates (x,y) on the graph and join them. The curves thus obtained is the graph of given inverse trig function.

Introduction to standard deviation

Standard deviation

The most comprehensive descriptions of dispersion are those that deal with the average deviation from some measure of central tendency. Two of these measures are important to our study of statistics: the variance and the standard deviation. Both of these tell us an average distance of any observation in the data set from the mean of the distribution.

Standard deviation definition
Earlier, when we calculated the range, the answers were expressed in the same units as the data. For the variance, however, the units are the squares of the units of the data – for example, “squared dollars” or “dollars squared”. Squared dollars or dollars squared are not intuitively clear or easily interpreted. For this reason, we have to make a significant change in the variance to compute useful measure of deviation one that does not give us a problem with units of measure and thus is less confusing. According to the , definition of standard deviation, this measure is called the standard deviation, and it is the square root of the variance. The square root of 100 dollars squared is 10 dollars because we take the square root of both the value and the units in which it is measured. The standard deviation, then, is in units that are the same as the original data.

Let us now define standard deviation: The standard deviation is a numerical measure of the average variability of a data set around its mean. The standard deviation for a population is denoted by s (Greek lower case letter sigma) and standard deviation for a sample denoted by s.The mean deviation has a limitation that it ignores the sign of x -   in the general case of an observation x. The standard deviation gets over this limitation by squaring x -  . (x -  )2 is positive whether or not x -   is negative or positive. Note that population means data set comprising of 100% items under study, and sample means data set comprising of sample items drawn out of the population so that by studying the sample, inferences may be drawn about the population.

Sequences and Series- An introduction

Sequence:


Let us consider the following collection of numbers – 
(1) 28,2,25,27,--------
(2) 2,7,11,19,31,51, ---------
(3) 1,2,3,4,5,6, --------------
(4) 20.5,18.5,16.5,14.5,12.5,10.5, ------------
In (1) the numbers are not arranged in a particular order. In (2) the numbers are in ascending order but they do not obey any rule or law. It is, therefore not possible to indicate the number next to 51.
In (3) we find that by adding 1 to any number, we get the next number. So the number after 6 would be = 6+1 = 7.
In (4) if we subtract 2 from any number we get the number that follows. So here the number after 10.5 would be = 10.5-2 = 8.5
Under these circumstances, we say, the numbers in the collections (1) and (2) do not form sequences whereas the numbers in the collections (3) and (4) form sequences.
Definition: An ordered collection of numbers a1,a2,a3,a4,….. an,…… is called a sequence if according to some definite rule or law there is a definite value of an, called the term or element of the sequence, corresponding to any value of the natural number n.
Clearly the nth term of a sequence is a function of the positive integer n. If the nth term itself is also always an integer, then such a sequence is called an integer sequence.

Series:

An expression of the form a1+a2+a3+a4+….. +an+….. which is the sum of the elements of a sequence {an} is called a series.

If Sn = u1+u2+u3+….un, the Sn is called the sum to n terms (or the sum of first n terms) of the series and is denoted by the Greek letter sign ?.
Thus Sn = ?_(r=1)^n¦u_r

If a sequence or a series contains finite number of elements, it is called a finite sequence or series, otherwise they are called infinite sequences and series.

Progressions:

There are mainly three types of progressions –
(a) Arithmetic progression (A.P.): That means a sequence in which each term is obtained by adding a constant d to the preceding term. This constant ‘d’ is called the common difference of the arithmetic progression.
(b) Geometric progression (G.P.): If in a sequence of terms each term is constant multiple of the proceeding term, then the sequence is called a Geometric Progression (G.P.). The constant  multiplier is called the common ratio (r).
(c) Harmonic progression (H.P.): If each term of an A.P. is replaced by its reciprocal, then we get a harmonic progression.

type of polygons

Few names of polygons are as follows:

• Triangle or Trigon

This is a three-sided shape that having whole interior angle measurement as 180º.

• Quadrilateral or Tetragon

This is a four-sided shape that having whole inside angle measurement as 360º.

• Pentagon

This is a five-sided shape which contains the whole inside angle measurement as 540º.
The polygon is a closed path. The polygon has the different shapes. The shapes are depends on the number of sides. The Straight lines are form the polygon shapes. The polygons are having more number of straight lines to form the different polygons. Angles of the polygons are varied based on shape of the polygon.
The area of the regular polygon can be written as,
Area = S² N / 4 tan (pi / N).
Here, S = Length of any side, N = Number of side, Pi = 3.14.

Area of the Polygons

1) The given length of side:
Area Shape of polygon = S² N / 4 tan (Pi/N)
Here, S = Side length, N = Number of side, Pi = 3.14.
2) Given the radius(circum radius):
Area shape of polygon = (R² N sin (2Pi/N))/2.
Here, R = Radius of the polygon, N = Number of side, Pi = 3.14.
3) Given the apothem (In radius):
Area Shape of polygon = A² N tan (Pi/N).
Here, A = Apothem length, N = Number of sides.
4) Given the apothem and Length of a side:
Area shape of polygon = A* P / 2.
Here, A = Apothem length, P = Perimeter.

More about Polygon:

Basic diagrammatic representation for polygon is as follows:

Polygon- A polygon is a closed figure made by joining line segments, where each line segment intersects exactly two others.

Regular Polygon- A regular polygon is a polygon whose sides are all the same length, and whose angles are all the same.

Quadrilateral- A four-sided polygon. The sum of the angles of a quadrilateral is 360^o.

Rectangle- A four-sided polygon having all right angles. The sum of the angles of a rectangle is 360^o.

Square- A four-sided polygon having equal-length sides meeting at right angles. The sum of the angles of a square is 360^o.

Parallelogram- A four-sided polygon with two pairs of parallel sides. The sum of the angles of a parallelogram is 360^o.

Rhombus- A four-sided polygon having all four sides of equal length. The sum of the angles of a rhombus is 360^o.

Trapezoid- A four-sided polygon having exactly one pair of parallel sides. The two sides that are parallel are called the bases of the trapezoid. The sum of the angles of a trapezoid is 360^o.

Pentagon- A five-sided polygon. The sum of the angles of a pentagon is 540^o.

Hexagon- A six-sided polygon. The sum of the angles of a hexagon is 720^o.

Octagon- An eight-sided polygon. The sum of the angles of an octagon is 1080^o.

Nonagon- A nine-sided polygon. The sum of the angles of a nonagon is 1260^o.

Decagon- A ten-sided polygon. The sum of the angles of a decagon is 1440^o.

Convex- A figure is convex if every line segment drawn between any two points inside the figure lies entirely inside the figure. A figure that is not convex is called a concave figure.

Area of a square = side x side

= s x s

= s² sq units or units²

Area of a Rectangle = length x width

= l x w

= lw sq units or units²

Area of a Parallelogram= base x height

= b x h

= bh sq units or units²

Area of a Triangle = ½ x base x height

= ½ b h sq units or units²

Area of a Rhombus = base x height

= b x h

= bh sq units or unit²

Area of a Trapezoid = ½ (a+b)h sq units or units²

(Half of the sum of the lengths of the

Parallel sides times height)