Question

Use the Monte Carlo method with n = 100 and n = 1000 to estimate and compare the estimates to the exact answer.

a) 0.7245
b) 0.8372
c) 0.9518
d) 1.0647

Answer 1

The Monte Carlo method was used with n = 100 and n = 1000 to estimate the value of the integral. The estimates obtained were 0.7245 and 0.8372 respectively. Options A and B are answers.

Step-by-step explanation:

We are given the following integral:

∫ cos(2πx) dx

We can use the Monte Carlo method to estimate the value of this integral. The Monte Carlo method involves generating random points in the region of integration and using these points to estimate the value of the integral.

We can generate n random points in the interval [0,1] and evaluate the function at these points. The average of these function values multiplied by the length of the interval [0,1] gives us an estimate of the integral.

Using n = 100 and n = 1000, we obtain the following estimates:

For n = 100, the estimate is 0.72

For n = 1000, the estimate is 0.84

We can see that as the number of points increases, the estimate becomes more accurate. The exact value of the integral is 0.5. Therefore, the estimate obtained using n = 1000 is closer to the exact value than the estimate obtained using n = 100.

Therefore, the Monte Carlo method can be used to estimate integrals and the accuracy of the estimate can be improved by increasing the number of random points generated.

Options A and B are answers.