1. Numerical and Geometric Integrals
7.3 Part I
Numerical and Geometric Integrals

2. I. Definite Integrals
The left and right integrals can be written with sigma notation:

3. II. Definite Integrals as Area

4. II. Definite Integrals as Area
B. You can use the calculator to estimate integrals using 𝑓𝑛𝐼𝑛𝑡
𝑓𝑛𝐼𝑛𝑡(𝑓 𝑥 ,𝑥,𝑎,𝑏)
C. Example 1: Estimate the following integrals using your calculator. Round to the nearest thousandth
𝑥 𝑥 𝑑𝑥
2.050
𝜋 𝑐𝑜𝑠𝑥𝑑𝑥
3. − 𝜋 𝑒 − 𝑥 𝑑𝑥
0.683
4. − 𝜋 𝑒 − 𝑥 𝑑𝑥
0.954
5. − 𝜋 𝑒 − 𝑥 𝑑𝑥
0.997

5. II. Definite Integrals as Area
C. Example 2: Evaluate the integral −1 1 1− 𝑥 2 𝑑𝑥 . 1. We see it is the semicircle pictured to the right −1 1 1− 𝑥 2 𝑑𝑥 = 1 2 𝜋 ∙1 2 −1 1 1− 𝑥 2 𝑑𝑥 = 𝜋 2 −1 1 1− 𝑥 2 𝑑𝑥 ≈ Estimate the value with your calculator −1 1 1− 𝑥 2 𝑑𝑥 ≈𝑓𝑛𝐼𝑛𝑡 1− 𝑥 2 ,𝑥,−1,1 −1 1 1− 𝑥 2 𝑑𝑥 ≈

6. III. Practice
Practice: Evaluate the following integrals using geometry (sketch a graph first). Check your answers with fnint() on your calculator.

7. III. Practice
a 𝑥+1 𝑑𝑥 𝐴𝑟𝑒𝑎= 1 2 ℎ 𝑏 1 + 𝑏 2 𝐴𝑟𝑒𝑎= 𝐴𝑟𝑒𝑎= 𝑥+1 𝑑𝑥=𝑓𝑛𝑖𝑛𝑡 𝑥 2 +1,𝑥,0,3 =5.25

8. III. Practice
b. 𝐴𝑟𝑒𝑎= 𝜋 𝑟 2 2 𝐴𝑟𝑒𝑎= 𝜋 =2𝜋 −2 2 4− 𝑥 2 𝑑𝑥 =𝑓𝑛𝑖𝑛𝑡 4− 𝑥 2 ,𝑥,−2,2 ≈6.283

9. III. Practice
c 𝑥−5 𝑑𝑥 𝐴𝑟𝑒𝑎= 1 2 𝑏 1 ℎ 𝑏 2 ℎ 2 𝐴𝑟𝑒𝑎= 1 2 ∙5∙ ∙5∙5 𝐴𝑟𝑒𝑎= 𝑥−5 𝑑𝑥=𝑓𝑛𝑖𝑛𝑡 𝑎𝑏𝑠 𝑥−5 ,𝑥,0,10 =25

10. Homework
Complete Worksheet 1