# Elaborating some earlier programs

1. New version of the table of factorials program: write to a file instead of the screen.
2. New version of the program to compute the integral of the function f(x) = fourth root of 1 - x^2 on the interval [0,1]: it should declare f as a C function.
3. The integral of e^{x^2/2}
4. The series 1 + 1/2^3 + 1/3^3 + 1/4^3 + ...

## Problem 1.

Rewrite your factorial program so that it writes its output to a file using fopen and fprintf.

## Problem 2

Rewrite your program to compute the integral of the fourth root of 1 - x^2 on the interval [0,1] so that it uses a function declaration.

## Problem 3

Modify the previous program to compute the integral of e^{-x^2/2} on the interval [-1,1]. Aim for two decimals of accuracy and discuss the precision actually obtained. Make a sketch of the geometric figure whose area the integral represents.

## Problem 4

Write a program which computes the sum
```  f(1) + f(2) + \cdots + f(n)
```
where f is an arbitrary function delared as a C function and where the user specifies n. Use this program to compute an approximation to the sum
```   beta = 1 + 1/2^3 + 1/3^3 + 1/4^3 + ...
```
accurate to two decimal places. What value of n need to achieve this accuracy? Justify your answer.
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