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Integral Calculus Notes | Study Mathematical Models - Physics

Document Description: Integral Calculus for Physics 2022 is part of Mathematical Models preparation. The notes and questions for Integral Calculus have been prepared according to the Physics exam syllabus. Information about Integral Calculus covers topics like and Integral Calculus Example, for Physics 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Integral Calculus.

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Line, Surface, and Volume Integrals

(a) Line Integrals
A line integral is an expression of the form
Integral Calculus Notes | Study Mathematical Models - Physics 
Integral Calculus Notes | Study Mathematical Models - Physics
whereIntegral Calculus Notes | Study Mathematical Models - Physicsis a vector function, Integral Calculus Notes | Study Mathematical Models - Physics is the infinitesimal displacement vector and the integral is to be carried out along a prescribed path P from point a to point b. If the path in question forms a closed loop (that is, if b = a), put a circle on the integral sign:
Integral Calculus Notes | Study Mathematical Models - Physics 
At each point on the path we take the dot product of Integral Calculus Notes | Study Mathematical Models - Physics (evaluated at that point) with the displacement Integral Calculus Notes | Study Mathematical Models - Physics to the next point on the path. The most familiar example of a line integral is the work done by a forceIntegral Calculus Notes | Study Mathematical Models - Physics
Integral Calculus Notes | Study Mathematical Models - Physics
Ordinarily, the value of a line integral depends critically on the particular path taken from a to b, but there is an important special class of vector functions for which the line integral is independent of the path, and is determined entirely by the end points(A force that has this property is called conservative.)

Example 14: Calculate the line integral of the function Integral Calculus Notes | Study Mathematical Models - Physicsfrom the point a = (1, 1, 0) to the point b = (2, 2, 0), along the paths (1) and (2) as shown in figure. What is Integral Calculus Notes | Study Mathematical Models - Physics for the loop that goes from a to b along (1) and returns to a along (2)?

Integral Calculus Notes | Study Mathematical Models - Physics
Since Integral Calculus Notes | Study Mathematical Models - PhysicsPath (1) consists of two parts. Along the “horizontal” segment dy = dz = 0, so
(i) Integral Calculus Notes | Study Mathematical Models - Physics
On the “vertical” stretch dx = dz = 0, so
(ii)Integral Calculus Notes | Study Mathematical Models - Physics
By path (1), then,
Integral Calculus Notes | Study Mathematical Models - Physics
Meanwhile, on path (2) x = y,  dx = dy, and dz = 0, so
Integral Calculus Notes | Study Mathematical Models - Physics
so
Integral Calculus Notes | Study Mathematical Models - Physics
For the loop that goes out (1) and back (2), then,
Integral Calculus Notes | Study Mathematical Models - Physics


Example 15: Find the line integral of the vector Integral Calculus Notes | Study Mathematical Models - Physics around a square of side ‘b’ which has a corner at the origin, one side on the x axis and the other side on the y axis.
Integral Calculus Notes | Study Mathematical Models - Physics

In a Cartesian coordinate systeIntegral Calculus Notes | Study Mathematical Models - Physics
Integral Calculus Notes | Study Mathematical Models - Physics
Along OP, y = 0, dy = 0 ⇒Integral Calculus Notes | Study Mathematical Models - Physics 
Along PQ , x = b,  dx = 0 ⇒ Integral Calculus Notes | Study Mathematical Models - Physics
Along QR, y = b, dy = 0 ⇒ Integral Calculus Notes | Study Mathematical Models - Physics
Along RO, x = 0,  dx = 0 ⇒ Integral Calculus Notes | Study Mathematical Models - Physics
Integral Calculus Notes | Study Mathematical Models - Physics


Example 16: Compute the line integralIntegral Calculus Notes | Study Mathematical Models - Physicsalong the triangular path shown in the figure.
Integral Calculus Notes | Study Mathematical Models - Physics

Line IntegralIntegral Calculus Notes | Study Mathematical Models - Physics
On path C1,  x = 0,  y = 0, Integral Calculus Notes | Study Mathematical Models - Physics
Integral Calculus Notes | Study Mathematical Models - Physics
On path C2, x = 0,  z = 0, Integral Calculus Notes | Study Mathematical Models - Physics
On path C3 the slope of line is -2 and intercept on z axis is 2 ⇒  z = -2y + 2 = 2 1 (1 - y) and the connecting points are (0, 1, 0) and (0, 0, 2)
On C3,  x=0, dx = 0 Integral Calculus Notes | Study Mathematical Models - Physics
Integral Calculus Notes | Study Mathematical Models - Physics
Integral Calculus Notes | Study Mathematical Models - Physics


Example 17: Given Integral Calculus Notes | Study Mathematical Models - Physicsin cylindrical coordinates. Find Integral Calculus Notes | Study Mathematical Models - Physics where c1 and c2 are contours shown in the figure.
Integral Calculus Notes | Study Mathematical Models - Physics

In cylindrical coordinate systemIntegral Calculus Notes | Study Mathematical Models - Physics
Integral Calculus Notes | Study Mathematical Models - Physics
In figure on curve c1,Ф varies from 0 to 2π, r = b and dr = 0
Integral Calculus Notes | Study Mathematical Models - Physics
On curve c2 , r = a,Ф varies from 0 to - 2π , and dr = 0 ⇒ Integral Calculus Notes | Study Mathematical Models - Physics
So, Integral Calculus Notes | Study Mathematical Models - Physics

(b) Surface Integrals
Integral Calculus Notes | Study Mathematical Models - Physics
A surface integral is an expression of the form
Integral Calculus Notes | Study Mathematical Models - Physics
where Integral Calculus Notes | Study Mathematical Models - Physics is again some vector function, and Integral Calculus Notes | Study Mathematical Models - Physics is an infinitesimal patch of area, with direction perpendicular to the surface(as shown in figure). There are, of course, two directions perpendicular to any surface, so the sign of a surface integral is intrinsically ambiguous. If the surface is closed then “outward” is positive, but for open surfaces it’s arbitrary.  If Integral Calculus Notes | Study Mathematical Models - Physics describes the flow of a fluid (mass per unit area per unit time), then Integral Calculus Notes | Study Mathematical Models - Physics arepresents the total mass per unit time passing through the surface-hence the alternative name, “flux.”
Ordinarily, the value of a surface integral depends on the particular surface chosen, but there is a special class of vector functions for which it is independent of the surface, and is determined entirely by the boundary line.

Example 18: Calculate the surface integral of A Integral Calculus Notes | Study Mathematical Models - Physics over five sides (excluding the bottom) of the cubical box (side 2) as shown in figure. Let “upward and outward” be the positive direction, as indicated by the arrows.

Taking the sides one at a time:
Integral Calculus Notes | Study Mathematical Models - Physics 
(i) Integral Calculus Notes | Study Mathematical Models - Physics
(ii)Integral Calculus Notes | Study Mathematical Models - Physics
(iii) Integral Calculus Notes | Study Mathematical Models - Physics
(iv) Integral Calculus Notes | Study Mathematical Models - Physics

(v)Integral Calculus Notes | Study Mathematical Models - Physics
Evidently the total flux is
Integral Calculus Notes | Study Mathematical Models - Physics 


Example 19:  Given a vector Integral Calculus Notes | Study Mathematical Models - Physics Evaluate Integral Calculus Notes | Study Mathematical Models - Physics  over the surface of the cube with the centre at the origin and length of side ‘a’.

The surface integral is performed on all faces. The differential surface on the different faces are Integral Calculus Notes | Study Mathematical Models - PhysicsFace abcd, Integral Calculus Notes | Study Mathematical Models - Physics
Integral Calculus Notes | Study Mathematical Models - PhysicsIntegral Calculus Notes | Study Mathematical Models - Physics
Face efgh, Integral Calculus Notes | Study Mathematical Models - Physics
Face cdfe, Integral Calculus Notes | Study Mathematical Models - Physics
Face aghb,Integral Calculus Notes | Study Mathematical Models - Physics
Similarly for the other two faces adfg and bceh we can find the surface integral withIntegral Calculus Notes | Study Mathematical Models - Physicsrespectively. The addition of these two surface integrals will be zero.
In the present case sum of all the surface integral
Integral Calculus Notes | Study Mathematical Models - Physics


Example 20: Use the cylindrical coordinate system to find the area of a curved surface on the right circular cylinder having radius = 3 m and height = 6 m and 30º << 120º.

From figure, surface area is required for a cylinder when r = 3m, z = 0 to 6m,
Integral Calculus Notes | Study Mathematical Models - Physics
Integral Calculus Notes | Study Mathematical Models - Physics
In cylindrical coordinate system, the elemental surface area as scalar isIntegral Calculus Notes | Study Mathematical Models - Physics
Taking the magnitude only
Integral Calculus Notes | Study Mathematical Models - Physics


Example 21: Use spherical coordinate system to find the area of the strip Integral Calculus Notes | Study Mathematical Models - Physics on the spherical shell of radius ‘a’. Calculate the area when α = 0 and β = π.
Integral Calculus Notes | Study Mathematical Models - Physics

Sphere has radius ‘a’ and θ varies between α and β.
For fixed radius the elemental surface is  da = (rsinθ d∅)(rdθ) = r2 sinθdθd∅
Integral Calculus Notes | Study Mathematical Models - Physics
For α = 0, β = π, Area = 2πa2 (1 + 1) = 4πa2 , is surface area of the sphere.


(c) Volume Integrals
A volume integral is an expression of the form
Integral Calculus Notes | Study Mathematical Models - Physics 
where T is a scalar function and dτ is an infinitesimal volume element. In Cartesian coordinates,
dτ = dx dy dz.
For example, if T is the density of a substance (which might vary from point to point) then the volume integral would give the total mass. Occasionally we shall encounter volume integrals of vector functions:
Integral Calculus Notes | Study Mathematical Models - Physics
because the unit vectors are constants, they come outside the integral.

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