Intro to Definite integrals Chapter 3.3

Lower limit of interval NOTATION Upper limit of interval 𝑎 𝑏 𝑓′ 𝑥 𝑑𝑥 Summation – remember that an integral represents the area under the curve of a derivative. It is the sum of several small intervals Equation of derivative – this does not necessarily imply first derivative. IT MUST BE KEPT IN CONTEXT. Lower limit of interval

−2 4 𝑥 2 +2𝑥 𝑑𝑥 NOTATION Example: −2 4 𝑥 2 +2𝑥 𝑑𝑥 This integral represents the area under the curve of 𝑥 2 +2𝑥 over the interval [-2, 4]

AREAS USING COMMON GEOMETRIC SHAPES Often the most accurate way to estimate the area under a curve is to use area formulas of common geometric shapes if the area can be divided in that way. Because it is more accurate, this method can be used to approximate a definite integral. 𝐸𝑣𝑎𝑢𝑙𝑎𝑡𝑒 0 4 𝑥𝑑𝑥 Since the area forms a triangle, use the area of a triangle formula to estimate the integral: 1 2 b h = 1 2 4 4 =16

Positive and Negative Area Areas that fall below the x-axis are considered negative. Areas that fall above the x- axis are considered positive. 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 −2 4 𝑥𝑑𝑥 Area from -2 to 0: 1 2 𝑏ℎ= 1 2 2 2 =2 Area from 0 to 4: 1 2 𝑏ℎ= 1 2 4 4 =4 Since the area from -2 to 0 is below the x-axis, it would be treated as negative. To estimate the definite integral, add the two areas: −2 4 𝑥𝑑𝑥 ≈ −2 +4=2

Using a calculator to evaluate definite integrals An exact value of the definite integral can be found using both algebraic and calculator methods. We will discuss algebraic methods later. To evaluate a definite integral using a calculuator: TI-84 without math type: - Type the function into y1 Press the math key and select 9 (fnInt) Follow the following format: fnInt(Y1, lower limit, upper limit, X) Press Enter TI-84 with math type: Type the function into Y1 The format will look just like the standard integral notation. Type the lower limit at the bottom of the integral symbol, the upper limit at the top of the integral symbol, and Y1 in the box next to the integral symbol. In the box next to the d, type x.

Using a calculator to evaluate definite integrals TI-NSPIRE: Press A to enter the Calculator app Press MENU CALCULUS INTEGRAL The format will look just like the standard integral notation. Type the lower limit at the bottom of the integral symbol, the upper limit at the top of the integral symbol, and the function in the box next to the integral symbol. In the box next to the d, type x. EXAMPLE: Use your calculator to evaluate −3 5 𝑋 4 +2 𝑥 2 +3 𝑑𝑥 ANSWER: If you did it correctly you should get approximately 798.93333 → →