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
- Integration-Definite
Exit problems of the session
-
Evaluate the integral using area formula: $\displaystyle\int_{-3}^3 \sqrt{9-x^2} dx$.
- 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 $f(x)$, $x-axis$, $x=a$, 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
- $\rhd$ 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)