M.Phil Physics
Course: Computational Physics (Stochastic Methods)
Home Work # 1
Due date: One Week after this distribution in class.
Question 1
Using Monte Carlo method, estimate the
shaded area as shown in figure.
(a) First find the value of area for
different number of darts thrown
(N). Then draw a graph for the
area versus N ( Take 1000 ≤ N ≤
106 in steps ).
(b) Then find out the exact value of
the area and find out error as
exact – calculated value. Plot the
error versus N. Comment on
results.
Question 2
The Normal distribution is given as p(x) = Aexp(-x2/2); A is given as 1/√2π.
Using Monte Carlo method, estimate the following:
(a) find the value of area under the curve for 0 ≤ x ≤ 1 for different number of
darts thrown (N). Then draw a graph for the area versus N.
(b) Then find out the exact value of the area and find out error as exact –
calculated value. Plot the error versus N. Comment on results.
Question 3
Using Monte Carlo method, estimate the area bounded by a parabola y = x2 and a
straight line y = x + 2.
(c) find the value of area under the curve for different number of darts
thrown (N). Then draw a graph for the area versus N.
(d) Then find out the exact value of the area and find out error as exact –
calculated value. Plot the error versus N. Comment on results.
Question 4
A cylinder has a diameter of 2 units and height of 2 units.
Using Monte Carlo method, estimate the following:
(e) find the value of volume for different number of darts thrown
(N). Then draw a graph for the area versus N.
(f) Then find out the exact value of the area and find out error as
exact – calculated value. Plot the error versus N. Comment on
results.
-1 +1
-1
+1 y = x
y = -x
Docsity.com
