1. Question

Solve the following Integrals,

1. ∫cos⁷x sin⁴x dx

2. ∫cos⁵x sin⁶x dx

2. Question

a. A circle with co-ordinates (0,2), (0,-2), (2,0), (-2,0) is rotated 360 degrees about the y-axis, what is the volume of the solid formed by this rotation.

b. Rotate a curve defined by the function y = x², between x = 2 and x = 4, about the,

i. x - axis

ii. y - axis

3. Question

Solve the integral U(x) = ∫tanθdθ.

4. Question

Solve the integral ∫cos(ln(x)).

5. Question

Two stiff parallel wires a distance d apart in a horizontal plane act as rails to support a light metal rod of mass m (perpendicular to each rail). A magnetic field ^→B, directed vertically upward acts throughout.

At t = 0, a constant current I begins to flow through the system. Determine the speed of the rod, which starts from rest at t = 0, as a function of time

(a) assuming no friction between the rod and the rails,

(b) if the coefficient of friction is μk

(c) In which direction does the rod move, east or west, if the current through it heads north?

6. Question

Write a program which takes in an input(number of alphabets) and prints out the alphabets such that a diamond shape is created at the core of the arrangement of the alphabets.

For example, if you enter 6, the program takes the alphabets A,B,C,D, E and F and arranges them in the manner shown below.

ABCDEFEDCBA
ABCDE EDCBA
ABCD    DCBA
ABC        CBA
AB            BA
A               A
AB            BA
ABC        CBA
ABCD    DCBA
ABCDE EDCBA
ABCDEFEDCBA