Integrals

Definite Integral


  • If we construct the graph of the integrand y = f(x), then in the case f(x) ≥ 0 the integral will be numerically equal to the area bounded by the curve y = f(x), the axis of x and the straight lines x = a and x = b.
  • Also \int_{a}^{b} f(x)dx=\mid F(x)\mid^{^b}_{_a}=F(b)-f(a), where F(x) is an anti derivative of f(x). If f(x) ≥ g(x) on [a, b], then the area bounded by y = f(x) and y = g(x) and the lines x = a and x = b is \int_{a}^{b} [f(x)-g(x)]dx.

View the Topic in this video From 10:54 To 43:18

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  • \int_{a}^{b} f(x)dx=\lim_{h \rightarrow 0}h[f(a)+f(a+h)+...+f(a+(n-1)h]\ 

     ⇒  \ \int_{a}^{b} f(x) dx=(b-a)\lim_{n \rightarrow \infty}\frac{1}{n}[f(a)+f(a+h)+...+f(a+(n-1)h)]\

                   where\ h=\frac{b-a}{n}\rightarrow 0\ as\ n\rightarrow \infty

  • \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{r=0}^{n-1} f \left(\frac{r}{n}\right) = \int_{o}^{1} f(x) dx
  • \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{r=1}^{n} f \left(\frac{r}{n}\right) = \int_{o}^{1} f(x) dx
  • \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{r=0}^{a(n-1)} f \left(\frac{r}{n}\right) = \int_{o}^{a} f(x) dx
  • \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{r=1}^{an} f \left(\frac{r}{n}\right) = \int_{o}^{a} f(x) dx