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Math  /  Calculus

QuestionUse a calculator or a computer to evaluate the definite integral.
Round your answer to four decimal places. ∫−55(e−t2)dt=\int_{-5}^{5}\left(e^{-t^{2}}\right) d t=

Studdy Solution

STEP 1

1. The integral ∫−55e−t2 dt\int_{-5}^{5} e^{-t^2} \, dt is a definite integral.
2. The function e−t2e^{-t^2} does not have an elementary antiderivative.
3. Numerical methods or computational tools are required to evaluate the integral.

STEP 2

1. Understand the nature of the integral.
2. Use a numerical method or computational tool to evaluate the integral.
3. Round the result to four decimal places.

STEP 3

Understand that the function e−t2e^{-t^2} is related to the Gaussian function, which is commonly encountered in probability and statistics. The integral over the entire real line is related to the error function, but here we are evaluating it over a finite interval.

STEP 4

Use a numerical integration method, such as Simpson's Rule, Trapezoidal Rule, or a computational tool like a calculator or software (e.g., Python, MATLAB, or Mathematica) to evaluate the definite integral ∫−55e−t2 dt\int_{-5}^{5} e^{-t^2} \, dt.
For example, using Python's `scipy.integrate.quad` function:
```python from scipy.integrate import quad import numpy as np
result, error = quad(lambda t: np.exp(-t**2), -5, 5) ```

STEP 5

Round the result obtained from the numerical evaluation to four decimal places. If using the Python example, the result would be stored in the variable `result`.
Assuming the result is approximately 1.7725, round it to four decimal places:
Rounded Result=1.7725 \text{Rounded Result} = 1.7725
The value of the definite integral is approximately:
1.7725 \boxed{1.7725}

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