Definite Integral Properties

Definite integrals are key in Calculus I, representing the area under curves. They connect to the Fundamental Theorem of Calculus, linking integration and differentiation, and offer various properties that simplify calculations and enhance understanding of functions over specified intervals.

1. Definite integral as the area under a curve

   • Represents the net area between the curve of a function and the x-axis over a specified interval.
   • Calculated as the limit of Riemann sums as the number of subintervals approaches infinity.
   • Can yield positive, negative, or zero values depending on the position of the curve relative to the x-axis.

2. Fundamental Theorem of Calculus

   • Connects differentiation and integration, showing that they are inverse processes.
   • States that if ( F ) is an antiderivative of ( f ), then ( \int_a^b f(x) , dx = F(b) - F(a) ).
   • Provides a method to evaluate definite integrals using antiderivatives.

3. Additivity of intervals

   • States that the integral over a combined interval can be expressed as the sum of integrals over subintervals: ( \int_a^c f(x) , dx = \int_a^b f(x) , dx + \int_b^c f(x) , dx ).
   • Useful for breaking down complex integrals into simpler parts.
   • Holds true regardless of whether the function is continuous or not.

4. Reversing integration limits

   • If the limits of integration are reversed, the integral changes sign: ( \int_a^b f(x) , dx = -\int_b^a f(x) , dx ).
   • This property is essential for understanding the orientation of area calculations.
   • Helps in simplifying integrals when the limits are not in the standard order.

5. Constant multiple rule

   • States that a constant can be factored out of the integral: ( \int_a^b k \cdot f(x) , dx = k \cdot \int_a^b f(x) , dx ).
   • Simplifies calculations when dealing with functions multiplied by constants.
   • Allows for easier manipulation of integrals in problem-solving.

6. Integration of even and odd functions

   • For even functions ( f(-x) = f(x) ), the integral over symmetric limits around zero simplifies to ( 2 \int_0^a f(x) , dx ).
   • For odd functions ( f(-x) = -f(x) ), the integral over symmetric limits is zero: ( \int_{-a}^a f(x) , dx = 0 ).
   • This property aids in quickly evaluating integrals without full computation.

7. Comparison theorem

   • Provides a method to estimate the value of an integral by comparing it to another integral.
   • If ( 0 \leq f(x) \leq g(x) ) for all ( x ) in ([a, b]), then ( \int_a^b f(x) , dx \leq \int_a^b g(x) , dx ).
   • Useful for determining convergence or divergence of improper integrals.

8. Average value of a function

   • The average value of a continuous function ( f ) over the interval ([a, b]) is given by ( \frac{1}{b-a} \int_a^b f(x) , dx ).
   • Provides a single value that represents the "central" tendency of the function over the interval.
   • Important for applications in physics and statistics.

9. Integration by substitution for definite integrals

   • A technique used to simplify integrals by changing variables: if ( u = g(x) ), then ( du = g'(x) , dx ).
   • Adjusts the limits of integration according to the substitution: if ( x = a ) then ( u = g(a) ) and if ( x = b ) then ( u = g(b) ).
   • Facilitates the evaluation of integrals that are otherwise difficult to compute.

10. Definite integrals of periodic functions

    • For periodic functions, the integral over one complete period can be used to find the integral over any multiple of that period.
    • The average value over one period can be multiplied by the number of periods to find the total integral.
    • Useful in applications involving oscillatory behavior, such as in physics and engineering.