Learning Outcomes

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

Textbook

  • 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)