- #1
TylerH
- 729
- 0
Given that 1/x is symetric across y=x, why can't we say [tex]\int^1_0 1/x - x dx= \int^\infty_1 1/x + x dx[/tex]? Geometrically, it makes sense, but ln(0) is clearly undefined.
The definite integral is a mathematical concept used to calculate the area under a curve on a specified interval. It is represented by the symbol ∫ and is the limit of the sum of infinitely small rectangles under the curve as the width of the rectangles approaches zero.
To solve this definite integral, we must first set up the integral using the limit definition. This means taking the limit as the width of the rectangles approaches zero and the number of rectangles approaches infinity. Then, we can use the properties of logarithms to simplify the integral and evaluate it using the Fundamental Theorem of Calculus.
The lower and upper limits of a definite integral represent the starting and ending points on the x-axis for the area under the curve that is being calculated. These limits determine the interval over which the integral is being evaluated and can greatly affect the value of the integral.
Yes, this integral can also be solved using the substitution method or integration by parts. However, the limit definition is the most commonly used method for evaluating definite integrals.
This integral can be used to solve problems involving exponential or logarithmic growth, such as calculating compound interest or population growth. It can also be applied in physics to calculate the work done by a variable force over a given distance.