Numerical and Geometric Integrals 7.3 Part I Numerical and Geometric Integrals

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

II. Definite Integrals as Area

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 1. 1 2 𝑥 𝑥 𝑑𝑥 2.050 2. 0 2𝜋 𝑐𝑜𝑠𝑥𝑑𝑥 3. −1 1 1 2𝜋 𝑒 − 𝑥 2 2 𝑑𝑥 0.683 4. −2 2 1 2𝜋 𝑒 − 𝑥 2 2 𝑑𝑥 0.954 5. −3 3 1 2𝜋 𝑒 − 𝑥 2 2 𝑑𝑥 0.997

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 ≈1.570796327 2. 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 ≈1.570796729

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

III. Practice a. 0 3 1 2 𝑥+1 𝑑𝑥 𝐴𝑟𝑒𝑎= 1 2 ℎ 𝑏 1 + 𝑏 2 𝐴𝑟𝑒𝑎= 1 2 3 1+ 5 2 𝐴𝑟𝑒𝑎= 21 4 0 3 1 2 𝑥+1 𝑑𝑥=𝑓𝑛𝑖𝑛𝑡 𝑥 2 +1,𝑥,0,3 =5.25

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

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

Homework Complete Worksheet 1