Learning Outcomes

  1. Understand the relationship between the definite integral and net area.
  2. Use geometry and properties of definite integrals to evaluate them. 


  • Chapter 5.2  The Definite Integral 

Textbook Assignment

  • p. 545:    72, 73, 76, 77, 80, 81, 88, 89, 91, 93

WeBWork Assignment

  • Integration-Definite

Exit problems of the session 

  1. Evaluate the integral using area formula: $\displaystyle\int_{-3}^3 \sqrt{9-x^2} dx$. 

  2. Suppose that $\displaystyle\int_{0}^4 f(x) dx=5$, $\displaystyle\int_{0}^2 f(x) dx=-1$, and $\displaystyle\int_{0}^4 g(x) dx=-2$, $\displaystyle\int_{0}^2 g(x) dx=3$, find 

$\displaystyle\int_{2}^4 [3f(x)+2g(x)]  dx$ .


Key Concepts

  • The definite integral is defined as the limit of a Riemann sum:

$\displaystyle\int_a^b f(x) dx=\displaystyle\lim_{n\to \infty}\sum_{i=1}^n f(x^*_i)\Delta x$ .

It measures the net signed area of the region enclosed by $f(x)$, $x-axis$, $x=a$, and $x=b$. 

  • Properties of Definite Integral:

$\displaystyle\int_a^a f(x) dx=0$

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

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

$\displaystyle\int_a^b cf(x) dx=c\displaystyle\int_a^b f(x) dx$      for constant $c$

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

  • Some definite integral can be evaluated by using areas of simple shapes, such as triangles, rectangles and circles. 


Videos and Practice Problems of Selected Topics

  1. $\rhd$ Definite integral (4:51) Defining $\displaystyle\int_a^bf(x)dx$ using Riemann sum.
  2. $\rhd$ Definite integral (2:28) Definite integrals represent the area between the curve of a function $y=f(x)$ and the $x$-axis.
  3. $\rhd$ Rewriting the limit of a Riemann sum as a definite integral (6:34) Write $\displaystyle\lim_{n\to\infty}\sum_{i=1}^n\ln\left(2+\dfrac{5i}{n}\right)$ as a definite integral. (Optional)
  4. $\rhd$ Finding definite integrals using properties (2:08) Evaluate $\displaystyle\int_3^3 f(x)dx$ and $\displaystyle\int_7^4 f(x)dx$ using graphs. 
  5. * Practice: Finding definite integral using properties. (4 problems)
  6. $\rhd$ Definite integral on adjacent intervals (3:05) Explain the property: $\displaystyle\int_a^b f(x) dx=\displaystyle\int_a^c f(x) dx+\displaystyle\int_c^b f(x) dx$
  7. $\rhd$ Breaking up the integral’s interval (7:24) Evaluate definite integral with geometry.
  8. $\rhd$ Merging definite integrals over adjacent intervals (4:01) Evaluate definite integral using properties. 
  9. * Practice: Finding definite integral over adjacent intervals. (4 problems)