How to Master All the Properties of Definite Integrals

Here is a Guide to completely understand the Properties of Definite Integrals:

The properties of definite integrals include additivity over intervals, reversal of limits (changing the sign), linearity (splitting sums and factoring constants), and the integral of a constant. Other key properties include non-negativity for non-negative functions, comparison between functions, and the mean value theorem for continuous functions. Here are the key properties:

1. Additivity Over Intervals

\(\int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx = \int_{a}^{c} f(x) \, dx\)

• The integral from \(a\) to \(c\) can be split into the sum of integrals over adjacent intervals \([a, b]\) and \([b, c]\).

2. Reversal of Limits

\(\int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx\)

• If the limits of integration are reversed, the sign of the integral is inverted.

3. Integral of a Constant

\(\int_{a}^{b} c \, dx = c(b – a)\)

• The integral of a constant \(c\) over an interval \([a, b]\) is the constant times the length of the interval.

4. Linearity of the Integral

\(\int_{a}^{b} [f(x) + g(x)] \, dx = \int_{a}^{b} f(x) \, dx + \int_{a}^{b} g(x) \, dx\)

\(\int_{a}^{b} c f(x) \, dx = c \int_{a}^{b} f(x) \, dx\)

• You can break up integrals across sums and factor constants out of the integral.

5. Zero-Width Interval

\(\int_{a}^{a} f(x) \, dx = 0\)

• If the limits of integration are the same, the integral is zero because there is no area under the curve.

6. Comparison Property

If \((f(x) \leq g(x))\) for all \(x\) in \([a, b]\), then:
\(\int_{a}^{b} f(x) \, dx \leq \int_{a}^{b} g(x) \, dx\)

• The integral of a smaller function is less than or equal to the integral of a larger function over the same interval.

7. Non-Negative Function Property

If \((f(x) \geq 0)\) for all \(x\) in \(([a, b]\)), then:
\(\int_{a}^{b} f(x) \, dx \geq 0\)

• The integral of a non-negative function over an interval is non-negative.

8. Mean Value Theorem for Definite Integrals

If \((f(x))\) is continuous on \([a, b]\), then there exists a \(c)\) in \([a, b]\) such that:
\(\int_{a}^{b} f(x) \, dx = f(c)(b – a)\)

• This means the integral is equal to the value of the function at some point in the interval multiplied by the length of the interval.

Understanding the properties of definite integrals is essential for solving calculus problems efficiently. These properties simplify complex calculations, help in breaking down integrals, and provide valuable insights into the behavior of functions over intervals. Mastering these properties enables better problem-solving and a deeper grasp of integral calculus.