NumericalIntegral

SRM 187 · 2004-03-16 · by Rustyoldman

SRM 187 · 2004-03-16 · by Rustyoldman · Advanced Math, String Manipulation

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Class Name: NumericalIntegral

Return Type: String

Method Name: integrate

Arg Types: (double, double)

Problem Statement

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Given two real numbers x1 and x2, calculate an approximation to the integral of e^{-x^2} evaluated between the limits from x1 to x2, which is accurate to the nearest 0.00001. Return the answer in a String, as a fixed point number with exactly five digits to the right of the decimal point and exactly one digit to the left of the decimal point.

For example: x1 = -0.5 and x2 = 0.5 returns "0.92256"

Notes

e-x^2 can be calculated in C++ with exp(-x*x) in math.h. e-x^2 can be calculated in C# with Math.Exp(-x*x). The Math class is in the System namespace.e-x^2 can be calculated in Java with Math.exp(-x*x). e-x^2 can be calculated in Visual Basic with Exp(-x*x) in the System.Math namespace.
The integral of a function is the area inside the closed figure formed by (on the top) the function between the limits of x=x1 and x=x2, (on the sides) vertical line segments at x=x1 and x=x2, and (on the bottom) the portion of the x axis between x=x1 and x=x2. This is shown by the shaded area above (the graph shows the function we are integrating, e-x^2).
The integral of e-x^2 is known to have no closed form, so don't waste time looking in a table of integrals for an exact formula.
Because of the 2e-6 constraint, about 40% of randomly chosen x1 and x2 values will be too close to a possible rounding error and will be rejected. This is not an error. It gives you more room for numerical errors.

Constraints

x1 will be less than x2.
x2-x1 will be between 0.00001 and 1.00000 inclusive.
x1 will be between -10.0 and 10.0 inclusive.
x2 will be between -10.0 and 10.0 inclusive.
To avoid rounding errors the inputs x1 and x2 must be chosen so that the answer is not within 2e-6 of 0.000005 + a multiple of 0.00001

Examples

-0.5
0.5
Returns: "0.92256"

The example from above. This is the largest possible answer given the constraints of this problem.

0.0
0.1
Returns: "0.09967"

2.0
2.451
Returns: "0.00368"

-9.0001
-9.0
Returns: "0.00000"

Values are very small out here.

2.5
3.0
Returns: "0.00034"

14)

1.22
1.23
Returns: "0.00223"

This is where the derivative is highest.

Submissions are judged against all 23 archived test cases, of which 6 are shown here. Case numbers match the judge’s.

Coding Area

Language: C++17 · define a public class NumericalIntegral with a public method string integrate(double x1, double x2) · 23 test cases · 2 s / 256 MB per case