SUMMARY
The improper integral $\int_{-\infty}^{0} e^{-|x|} dx$ simplifies to $\int_{-\infty}^{0} e^{-x} dx$ because for $x < 0$, $|x| = -x$. The correct evaluation of this integral yields a result of 1, not -1. The confusion arises from misapplying the limits and the exponential function. The integral evaluates to $e^{-x}$, and upon solving, the limit as $x$ approaches $-\infty$ results in a total area of 1 under the curve.
PREREQUISITES
- Understanding of improper integrals
- Knowledge of exponential functions
- Familiarity with the properties of absolute values
- Basic calculus skills, specifically integration techniques
NEXT STEPS
- Study the evaluation of improper integrals in detail
- Learn about the properties of the exponential function, particularly $e^{-x}$
- Explore the concept of absolute values in piecewise functions
- Practice solving similar integrals with different limits and functions
USEFUL FOR
Students studying calculus, particularly those focusing on integration techniques, and educators looking for clarification on improper integrals.