Properties of Definite Integrals

Properties of Definite Integrals

  • Understanding Definite Integrals

    • Definite integrals measure the area under a curve from a starting point (lower limit) to an ending point (upper limit).
    • Example: The definite integral from 0 to 12 calculates the area under the curve between these points.

  • First Problem Example

    • Starting from an upper limit of 12 to a lower limit of 0, area calculation involves:
    • Finding the area above and below the x-axis.
    • For example:
      • Area = 2extpi+2+2+1−4=2extpi+12 ext{pi} + 2 + 2 + 1 - 4 = 2 ext{pi} + 1

  • Reversing Limits

    • If the limits are reversed (12 to 0 instead of 0 to 12), the area changes sign:
    • Example:
      • From positive contributions, it turns negative, resulting in:
      • Area = −1−2extpi-1 - 2 ext{pi}
    • Thus, swapping limits leads to the opposite sign result:
      • extArea(b,a)=−extArea(a,b)ext{Area}(b, a) = - ext{Area}(a, b)

  • Equivalent Limits

    • If the lower and upper limits are the same, the area is zero.
    • Example: extArea(a,a)=0ext{Area}(a, a) = 0

  • Multiplying by a Constant

    • When a function is multiplied by a constant, the constant can be factored out.
    • Rule:
      • extArea(cextf(x),a,b)=cextArea(extf(x),a,b)ext{Area}(c ext{f}(x), a, b) = c ext{Area}( ext{f}(x), a, b)

  • Additive Property of Integrals

    • Splitting up integrals is allowed if there is a point c between a and b.
    • Rule:
      • extArea(a,c)+extArea(c,b)=extArea(a,b)ext{Area}(a, c) + ext{Area}(c, b) = ext{Area}(a, b)

  • Function Addition and Subtraction

    • Two functions can be integrated separately:
    • extArea(a,b)(f+g)=extArea(a,b)(f)+extArea(a,b)(g)ext{Area}(a, b)(f+g) = ext{Area}(a, b)(f) + ext{Area}(a, b)(g)
    • The same applies for subtraction.

  • Understanding $d x$

    • The notation dxd x indicates the integral is calculated with respect to the x-axis.

  • Examples of Area Calculation

    • Consider ranges with known areas:
    • If from 7 to 6, it's negative 2 (moving left).
    • Adjustments using integral properties can provide answers involving given ranges.

  • Piecewise Functions

    • When calculating areas under piecewise functions, treat them as separate segments.
    • For example:
    • Combine the areas geometrically even if there are jumps.

  • Absolute Value in Integrals

    • When absolute values are involved, the integral can change significantly depending on whether the absolute value is computed before or after solving the integral.
    • Case 1: Using $ ext{Abs}$ after calculating integral gives areas as is.
    • Case 2: Using $ ext{Abs}$ before flips negative areas up to positive.

  • Calculating with a Calculator

    • For complicated integrals, the use of a calculator is optimal:
    • On a TI calculator, you can use math option 9 to integrate functions easily.
    • Example: for extintegral(extsqrt(x−1),2,3)ext{integral}( ext{sqrt}(x-1), 2, 3) to find the area under the curve from 2 to 3.