5.3 – The Definite Integral and the Fundamental Theorem of Calculus Math 140 5.3 – The Definite Integral and the Fundamental Theorem of Calculus

Q: How can we find the area under any curve Q: How can we find the area under any curve? For example, let’s try to find the area under 𝑓 𝑥 = 𝑥 3 −9 𝑥 2 +25𝑥−19

What is the area of this?

1 rectangle

2 rectangles

4 rectangles

50 rectangles

If the left sides of our rectangles are at 𝑥 1 , 𝑥 2 , …, 𝑥 𝑛 and each rectangle has width Δ𝑥, then the areas of the rectangles are 𝑓 𝑥 1 Δ𝑥, 𝑓 𝑥 2 Δ𝑥, …, 𝑓 𝑥 𝑛 Δ𝑥. So, the sum of the areas of the rectangles (called a Riemann sum) is: 𝑓 𝑥 1 Δ𝑥+𝑓 𝑥 2 Δ𝑥+…+𝑓 𝑥 𝑛 Δ𝑥 = 𝑓 𝑥 1 +𝑓 𝑥 2 +…+𝑓 𝑥 𝑛 Δ𝑥

So, the sum of the areas of the rectangles (called a Riemann sum) is: 𝑓 𝑥 1 Δ𝑥+𝑓 𝑥 2 Δ𝑥+…+𝑓 𝑥 𝑛 Δ𝑥 = 𝑓 𝑥 1 +𝑓 𝑥 2 +…+𝑓 𝑥 𝑛 Δ𝑥

So, the sum of the areas of the rectangles (called a Riemann sum) is: 𝑓 𝑥 1 Δ𝑥+𝑓 𝑥 2 Δ𝑥+…+𝑓 𝑥 𝑛 Δ𝑥 = 𝑓 𝑥 1 +𝑓 𝑥 2 +…+𝑓 𝑥 𝑛 Δ𝑥

As the number of rectangles (𝑛) increases, the above sum gets closer to the true area. That is, Area under curve= 𝒍𝒊𝒎 𝐧→+∞ 𝒇 𝒙 𝟏 +𝒇 𝒙 𝟐 +…+𝒇 𝒙 𝒏 𝜟𝒙

𝑎 𝑏 𝑓 𝑥 𝑑𝑥 = lim n→+∞ 𝑓 𝑥 1 +𝑓 𝑥 2 +…+𝑓 𝑥 𝑛 Δ𝑥 The above integral is called a definite integral, and 𝑎 and 𝑏 are called the lower and upper limits of integration.

The Fundamental Theorem of Calculus If 𝑓(𝑥) is continuous on the interval 𝑎≤𝑥≤𝑏, then 𝒂 𝒃 𝒇 𝒙 𝒅𝒙 =𝑭 𝒃 −𝑭(𝒂) where 𝐹(𝑥) is any antiderivative of 𝑓(𝑥) on 𝑎≤𝑥≤𝑏.

Ex 1. Evaluate the given definite integrals using the fundamental theorem of calculus. −1 1 2 𝑥 2 𝑑𝑥

Ex 1. Evaluate the given definite integrals using the fundamental theorem of calculus. −1 1 2 𝑥 2 𝑑𝑥

Ex 1 (cont). Evaluate the given definite integrals using the fundamental theorem of calculus. 0 1 ( 𝑒 2𝑥 − 𝑥 ) 𝑑𝑥

Ex 1 (cont). Evaluate the given definite integrals using the fundamental theorem of calculus. 0 1 ( 𝑒 2𝑥 − 𝑥 ) 𝑑𝑥

Note: Definite integrals can evaluate to negative numbers Note: Definite integrals can evaluate to negative numbers. For example, 1 2 (1−𝑥) 𝑑𝑥 =− 1 2

Just like with indefinite integrals, terms can be integrated separately, and constants can be pulled out. 𝑎 𝑏 𝑓 𝑥 ±𝑔(𝑥) 𝑑𝑥 = 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 ± 𝑎 𝑏 𝑔(𝑥) 𝑑𝑥 𝑎 𝑏 𝑘𝑓 𝑥 𝑑𝑥 =𝑘 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 Also, 𝑎 𝑎 𝑓 𝑥 𝑑𝑥 =0 𝑏 𝑎 𝑓 𝑥 𝑑𝑥 =− 𝑎 𝑏 𝑓 𝑥 𝑑𝑥

𝑎 𝑏 𝑓 𝑥 𝑑𝑥 = 𝑎 𝑐 𝑓 𝑥 𝑑𝑥 + 𝑐 𝑏 𝑓 𝑥 𝑑𝑥 (here, 𝑐 is any real number)

𝑎 𝑏 𝑓 𝑥 𝑑𝑥 = 𝑎 𝑐 𝑓 𝑥 𝑑𝑥 + 𝑐 𝑏 𝑓 𝑥 𝑑𝑥 (here, 𝑐 is any real number)

𝑎 𝑏 𝑓 𝑥 𝑑𝑥 = 𝑎 𝑐 𝑓 𝑥 𝑑𝑥 + 𝑐 𝑏 𝑓 𝑥 𝑑𝑥 (here, 𝑐 is any real number)

Ex 2. Let 𝑓(𝑥) and 𝑔(𝑥) be functions that are continuous on the interval −3≤𝑥≤5 and that satisfy −3 5 𝑓 𝑥 𝑑𝑥 =2 −3 5 𝑔 𝑥 𝑑𝑥 =7 1 5 𝑓 𝑥 𝑑𝑥 =−8 Use this information along with the rules definite integrals to evaluate the following. −3 5 2𝑓 𝑥 −3𝑔 𝑥 𝑑𝑥 −3 1 𝑓 𝑥 𝑑𝑥

Ex 3. Evaluate: 0 1 8𝑥 𝑥 2 +1 3 𝑑𝑥

Ex 3. Evaluate: 0 1 8𝑥 𝑥 2 +1 3 𝑑𝑥

Net Change If we have 𝑄′(𝑥), then we can calculate the net change in 𝑄(𝑥) as 𝑥 goes from 𝑎 to 𝑏 with an integral: 𝑄 𝑏 −𝑄 𝑎 = 𝑎 𝑏 𝑄 ′ 𝑥 𝑑𝑥

Net Change Ex 4. Suppose marginal cost is 𝐶 ′ 𝑥 =3 𝑥−4 2 dollars per unit when the level of production is 𝑥 units. By how much will the total manufacturing cost increase if the level of production is raised from 6 units to 10 units?