Definite Integrals

Definite integrals Starter: KUS objectives BAT evaluate definite integrals Starter: Find 3 𝑥 −2 𝑑𝑥 Find 𝑥− 1 𝑥 2 + 3 𝑥 𝑑𝑥 Find 𝑥 𝑥 − 3 𝑥 𝑥 𝑑𝑥

Then 𝑎 𝑏 𝑓 ′ 𝑥 𝑑𝑥 = 𝑓(𝑥) 𝑏 𝑎 =𝑓 𝑏 −𝑓(𝑎) Introduction If 𝑓 ′ 𝑥 𝑑𝑥=𝑓 𝑥 +𝐶 What happens to + C? Then 𝑎 𝑏 𝑓 ′ 𝑥 𝑑𝑥 = 𝑓(𝑥) 𝑏 𝑎 =𝑓 𝑏 −𝑓(𝑎) Note that 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 is called a definite integral since it gives a definite answer WB1 = 2 𝑥 2 +6𝑥 5 2 Find 2 5 4𝑥+6 𝑑𝑥 = 2 25 +30 −(2 4 +12) = 80 − 20 = 60

Evaluate the following: 1 2 3 𝑥 2 𝑑𝑥 WB2 Evaluate the following: 1 2 3 𝑥 2 𝑑𝑥 Your workings must be clear here. There are 3 stages… Integrate the function and put it in square brackets. Put the ‘limits’ outside the bracket. The statement. Basically the function written out, with values for a and b Split the integration into 2 separate brackets After integration. The function is integrated and put into square brackets Substitute ‘b’ into the first, and ‘a’ into the second The evaluation. Round brackets are used to split the integration in two. One part for b and one for a.

WB3 Evaluate the following: 1 4 2𝑥−3 𝑥 1 2 +1 𝑑𝑥 Integrate the function and put it in square brackets. Put the ‘limits’ outside the bracket. Simplify if possible Split and substitute

Split into 2 and substitute b and a WB4 Evaluate the following: −1 0 𝑥 1/3 −1 2 𝑑𝑥 𝑥 5 3 5 3 − 2𝑥 4 3 4 3 +𝑥 1 4 3 5 𝑥 5 3 − 3 2 𝑥 4 3 +𝑥 1 4 Split into 2 and substitute b and a

WB5 Evaluate these definite integrals: a) −1 3 𝑥+1)(𝑥−3 𝑑𝑥 b) 1 2 4 𝑥 2 𝑥+1 𝑑𝑥 c) 1 3 10𝑥−6 𝑥 2 𝑑𝑥 d) −2 −1 𝑥+2 2 𝑑𝑥 Solutions: a) b) c) −12 d) 1 3

Practice1 Evaluate these integrals = 𝑥 4 2 1 1 2 4 𝑥 3 𝑑𝑥 = 2 4 − 1 4 =15 1 3 3 𝑥 2 −1 𝑑𝑥 = 𝑥 3 −𝑥 3 1 = 27−3 − 1−1 =24 −1 2 4𝑥+1 𝑑𝑥 = 2𝑥 2 +𝑥 2 −1 = 8+2 − 2−1 =9

Practice 2

Practice 3

One thing to improve is – KUS objectives BAT evaluate definite integrals self-assess One thing learned is – One thing to improve is –

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