|TIME: 3 HOURS|
where R is the region in
between the curves y = x and
y = x4, for
where Da is the disc of radius a > 0,
centred at the origin.
where E is the region in
bounded by the
spheres of radii a and b (b > a > 0), centred at the origin.
||Show that the integral
where S is the unit square
||Compute an approximate value for I by partitioning S by the
and then using double sum with the
integrand evaluated at the mid-points of the sub-regions defined by
||Consider the vector field
defined on .
||Find a potential function for .
||Evaluate the line integral
is any smooth curve joining A to B.
||Compute the flux of
across the ellipsoid
b2y2 + c2z2 = 1.
||Use the change of coordinates in
where T is the triangular region in the (x,y)
plane whose vertices are
||In this problem you may use the standard
formulas for area of a disc and surface area of a sphere:
respectively, where R is the radius.
is the circle of radius 3 centred at the origin, oriented
||Compute the value of
is the sphere
x2 + y2 + z2 = a2, oriented by its
where e is a constant and
is the unit normal. Express e in terms of Q
||Suppose that two vector fields
and smooth on ,
are related by curl
Show that the flux
of J through a surface
is equal to the
is the boundary of .
are smooth and compatibly
||Determine which of the following sequences
converges and obtain the
limit in the convergent cases.
||Determine whether each of the following series is absolutely
convergent , conditionally convergent or divergent.
||(i)||Find either the general (n
or the terms of degree ,
Taylor series expansion about x = 0 (Maclaurin series) for
and determine the interval of convergence in each case.
||Set x = 1/3 in (a) (ii) and compute the approximation to
given by the sum of the first three non-zero terms of the Taylor series.
Compute the error, if the exact value is
||Compute an approximation to
by using the first three non-zero terms in the Taylor expansion for
f(t) about t = 0. Use this result to approximate F(1/2).