5.4 – Applying Definite Integration

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Presentation transcript:

5.4 – Applying Definite Integration Math 140 5.4 – Applying Definite Integration

Q: How can we find the area between two curves?

If 𝑓 and 𝑔 are continuous with 𝑓 𝑥 ≥𝑔(𝑥) on the interval 𝑎≤𝑥≤𝑏, then Area between curves = 𝑎 𝑏 𝑓 𝑥 −𝑔 𝑥 𝑑𝑥

Ex 1. Find the area of the region enclosed by the curves 𝑦= 𝑥 3 and 𝑦= 𝑥 2 .

Ex 1. Find the area of the region enclosed by the curves 𝑦= 𝑥 3 and 𝑦= 𝑥 2 .

Ex 2. Find the area of the region bounded by the line 𝑦=4𝑥 and the curve 𝑦= 𝑥 3 +3 𝑥 2 .

Ex 2. Find the area of the region bounded by the line 𝑦=4𝑥 and the curve 𝑦= 𝑥 3 +3 𝑥 2 .

Q: How do we find the average value of a continuous function Q: How do we find the average value of a continuous function? The average value of 𝑓(𝑥) over 𝑎≤𝑥≤𝑏 is: 𝑉= 1 𝑏−𝑎 𝑎 𝑏 𝑓 𝑥 𝑑𝑥

Q: How do we find the average value of a continuous function Q: How do we find the average value of a continuous function? The average value of 𝑓(𝑥) over 𝑎≤𝑥≤𝑏 is: 𝑉= 1 𝑏−𝑎 𝑎 𝑏 𝑓 𝑥 𝑑𝑥

Ex 3. A manufacturer supplies 𝑆 𝑝 =0 Ex 3. A manufacturer supplies 𝑆 𝑝 =0.3 𝑝 2 +6𝑝+14 thousand coasters to the market when the price is 𝑝 dollars per coaster. Find the average supply as the price varies from 𝑝=$2 to 𝑝=$5.