1. Exploring areas between curves Task Three

2. THE PROBLEM Find the area bounded by 𝑦=𝑥2+1 and 𝑦=−𝑥2+2𝑥+1. STEP 1: Graph the equations.

3. THE PROBLEM Find the area bounded by 𝑦=𝑥2+1 and 𝑦=−𝑥2+2𝑥+1. STEP 1: Graph the equations.

4. THE PROBLEM Find the area bounded by 𝑦=𝑥2+1 and 𝑦=−𝑥2+2𝑥+1. STEP 2: Find the intersections to determine the x values which bound the region of the unknown area. (1, 2) (0, 1)

5. THE PROBLEM Find the area bounded by 𝑦=𝑥2+1 and 𝑦=−𝑥2+2𝑥+1. STEP 3: Use logic to determine the best way to get the area. (1, 2) (0, 1) Area of Yellow = Area of Blue – Area of Purple within 0≤𝑥≤1

6. THE PROBLEM Find the area bounded by 𝑦=𝑥2+1 and 𝑦=−𝑥2+2𝑥+1. STEP 4: SOLVE!

7. THE SPECIAL PROPERTY Find the area bounded by 𝑦=𝑥2+1 and 𝑦=−𝑥2+2𝑥+1. 𝒂𝒃𝒇𝒙±𝒈𝒙𝒅𝒙=𝒂𝒃𝒇𝒙𝒅𝒙±𝒂𝒃𝒈𝒙𝒅𝒙 Area = (Area of Blue – Area of Purple) Area =𝑎𝑏𝑓𝑥𝑑𝑥−𝑎𝑏𝑔𝑥𝑑𝑥 Area =𝑎𝑏𝑓𝑥±𝑔𝑥𝑑𝑥 Area=𝑎𝑏((−𝑥2+2𝑥+1) − (𝑥2+1))𝑑𝑥 Area= 𝑎𝑏(−2𝑥2+2𝑥)𝑑𝑥

8. THE SPECIAL PROPERTY Find the area bounded by 𝑦=𝑥2+1 and 𝑦=−𝑥2+2𝑥+1. *CHECK IF ANSWERS ARE CONSISTENT. Area= 𝑎𝑏(−2𝑥2+2𝑥)𝑑𝑥