Integral Calculus IIT JAM Notes | EduRev

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IIT JAM : Integral Calculus IIT JAM Notes | EduRev

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Line, Surface, and Volume Integrals

(a) Line Integrals
A line integral is an expression of the form
Integral Calculus IIT JAM Notes | EduRev 
Integral Calculus IIT JAM Notes | EduRev
whereIntegral Calculus IIT JAM Notes | EduRevis a vector function, Integral Calculus IIT JAM Notes | EduRev 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 IIT JAM Notes | EduRev 
At each point on the path we take the dot product of Integral Calculus IIT JAM Notes | EduRev (evaluated at that point) with the displacement Integral Calculus IIT JAM Notes | EduRev to the next point on the path. The most familiar example of a line integral is the work done by a forceIntegral Calculus IIT JAM Notes | EduRev
Integral Calculus IIT JAM Notes | EduRev
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 IIT JAM Notes | EduRevfrom 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 IIT JAM Notes | EduRev for the loop that goes from a to b along (1) and returns to a along (2)?

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


Example 15: Find the line integral of the vector Integral Calculus IIT JAM Notes | EduRev 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 IIT JAM Notes | EduRev

In a Cartesian coordinate systeIntegral Calculus IIT JAM Notes | EduRev
Integral Calculus IIT JAM Notes | EduRev
Along OP, y = 0, dy = 0 ⇒Integral Calculus IIT JAM Notes | EduRev 
Along PQ , x = b,  dx = 0 ⇒ Integral Calculus IIT JAM Notes | EduRev
Along QR, y = b, dy = 0 ⇒ Integral Calculus IIT JAM Notes | EduRev
Along RO, x = 0,  dx = 0 ⇒ Integral Calculus IIT JAM Notes | EduRev
Integral Calculus IIT JAM Notes | EduRev


Example 16: Compute the line integralIntegral Calculus IIT JAM Notes | EduRevalong the triangular path shown in the figure.
Integral Calculus IIT JAM Notes | EduRev

Line IntegralIntegral Calculus IIT JAM Notes | EduRev
On path C1,  x = 0,  y = 0, Integral Calculus IIT JAM Notes | EduRev
Integral Calculus IIT JAM Notes | EduRev
On path C2, x = 0,  z = 0, Integral Calculus IIT JAM Notes | EduRev
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 IIT JAM Notes | EduRev
Integral Calculus IIT JAM Notes | EduRev
Integral Calculus IIT JAM Notes | EduRev


Example 17: Given Integral Calculus IIT JAM Notes | EduRevin cylindrical coordinates. Find Integral Calculus IIT JAM Notes | EduRev where c1 and c2 are contours shown in the figure.
Integral Calculus IIT JAM Notes | EduRev

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

(b) Surface Integrals
Integral Calculus IIT JAM Notes | EduRev
A surface integral is an expression of the form
Integral Calculus IIT JAM Notes | EduRev
where Integral Calculus IIT JAM Notes | EduRev is again some vector function, and Integral Calculus IIT JAM Notes | EduRev 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 IIT JAM Notes | EduRev describes the flow of a fluid (mass per unit area per unit time), then Integral Calculus IIT JAM Notes | EduRev 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 IIT JAM Notes | EduRev 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 IIT JAM Notes | EduRev 
(i) Integral Calculus IIT JAM Notes | EduRev
(ii)Integral Calculus IIT JAM Notes | EduRev
(iii) Integral Calculus IIT JAM Notes | EduRev
(iv) Integral Calculus IIT JAM Notes | EduRev

(v)Integral Calculus IIT JAM Notes | EduRev
Evidently the total flux is
Integral Calculus IIT JAM Notes | EduRev 


Example 19:  Given a vector Integral Calculus IIT JAM Notes | EduRev Evaluate Integral Calculus IIT JAM Notes | EduRev  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 IIT JAM Notes | EduRevFace abcd, Integral Calculus IIT JAM Notes | EduRev
Integral Calculus IIT JAM Notes | EduRevIntegral Calculus IIT JAM Notes | EduRev
Face efgh, Integral Calculus IIT JAM Notes | EduRev
Face cdfe, Integral Calculus IIT JAM Notes | EduRev
Face aghb,Integral Calculus IIT JAM Notes | EduRev
Similarly for the other two faces adfg and bceh we can find the surface integral withIntegral Calculus IIT JAM Notes | EduRevrespectively. The addition of these two surface integrals will be zero.
In the present case sum of all the surface integral
Integral Calculus IIT JAM Notes | EduRev


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 IIT JAM Notes | EduRev
Integral Calculus IIT JAM Notes | EduRev
In cylindrical coordinate system, the elemental surface area as scalar isIntegral Calculus IIT JAM Notes | EduRev
Taking the magnitude only
Integral Calculus IIT JAM Notes | EduRev


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

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 IIT JAM Notes | EduRev
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 IIT JAM Notes | EduRev 
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 IIT JAM Notes | EduRev
because the unit vectors are constants, they come outside the integral.

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