Solved problems on calculus, mechanics, differential equations, linear algebra and matrix

Solution 1: The given double integral is
Let , where
Now, consider
. Compare with , we get
.... (1)
And consider
. Compare with , we get
.... (2)
(a): The required region of integration from equation (1) and (2) is shown below.
(b): Now, take a strip parallel to x – axis, and go from left to right. The strip intersect
circle and then and , that is
So, by change of order of integration
Solution 2:

Let be any point on the plate. Since the pressure (force per unit area) at any point
on the plate is proportional to the square of the distance of that point from one corner.
That is
The total force F is
Solution 3: Given the force function
, and
This implies that
So,
And
The work done is
So, wok done is

Solution 4: Given that . Compare with P we get
Now,
, and .
Since, , this implies that the function F is a conservative force. Then there exist
a potential function such that
This implies that
Comparing both sides we get
Integrating equation (1) we get
Differentiate equation (3) with respect to y partially we get
Using equation (2) we get
Use g(y) in equation (3), the potential function is
Now, the work done is
Solution 5: Given the motion of the spring system with friction is described by the
differential equation
a): The characteristic equation is . Solving we get