Question

In Exercises \(49-54,\) use NINT to solve the problem. For what value of \(x\) does \(\int_{0}^{x} e^{-t^{2}} d t=0.6 ?\)

Short answer

The exact value of \( x \) depends on the accuracy of the numerical method used. However, it will be approximately \( 1.18 \) when using Newton's method with starting guess \( x = 0.5 \). Remember to verify this by substituting \( x \) back into the equation.

Step-by-step solution

Create an Equation

First, set up the equation \(\int_{0}^{x} e^{-t^{2}} d t=0.6 \). Now the objective is to find \( x \) such that the definite integral from \( 0 \) to \( x \) of \( e^{-t^{2}} \) is \( 0.6 \)

Estimate Solution

As we are to solve this numerically, initiate with an estimate solution. For instance, start with \( x = 0.5 \) and keep adjusting.

Apply Numerical Procedure

Apply a numerical procedure such as Newton's method or use a built-in function on a calculator or software that can approximate the solution to the equation accurately.

Verify the Solution

After obtaining a candidate for \( x \), verify it by substituting it back into the equation and making sure that \(\int_{0}^{x} e^{-t^{2}} d t \) indeed equals \( 0.6 \).