Learning Outcomes
 Understand the relationship between the definite integral and net area.
 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
 IntegrationDefinite
Exit problems of the session

Evaluate the integral using area formula: $\displaystyle\int_{3}^3 \sqrt{9x^2} dx$.
 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 $f(x)$, $xaxis$, $x=a$, 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
 $\rhd$ Definite integral (4:51) Defining $\displaystyle\int_a^bf(x)dx$ using Riemann sum.
 $\rhd$ Definite integral (2:28) Definite integrals represent the area between the curve of a function $y=f(x)$ and the $x$axis.
 $\rhd$ 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)
 $\rhd$ 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.
 * Practice: Finding definite integral using properties. (4 problems)
 $\rhd$ Definite integral on adjacent intervals (3:05) Explain the property: $\displaystyle\int_a^b f(x) dx=\displaystyle\int_a^c f(x) dx+\displaystyle\int_c^b f(x) dx$
 $\rhd$ Breaking up the integral’s interval (7:24) Evaluate definite integral with geometry.
 $\rhd$ Merging definite integrals over adjacent intervals (4:01) Evaluate definite integral using properties.
 * Practice: Finding definite integral over adjacent intervals. (4 problems)