6) Definite Integrals and properties, Definite Integral as the limit of a sum:

  • A definite integral is the area under a curve between two endpoints.
  • It can be calculated using the limit of a sum of rectangles that approximate the area under the curve.
  • The properties of definite integrals include linearity, additivity, and symmetry.

Important Formulas to Remember in these topic


$$\int_a^b f(x)dx = F(b) - F(a)$$


$$\int_a^b f(x)dx = -\int_b^a f(x)dx$$


$$\int_a^b [f(x) \pm g(x)]dx = \int_a^b f(x)dx \pm \int_a^b g(x)dx$$


$$\int_a^b kf(x)dx = k\int_a^b f(x)dx$$


$$\int_a^b f(x)dx + \int_b^c f(x)dx = \int_a^c f(x)dx$$


$$\int_a^b f(x)dx = \lim_{n\to\infty}\frac{b-a}{n}\sum_{i=1}^{n} f(x_i)$$