Areas between curves
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The document describes finding the area between two curves, y=x^2+1 and y=-x^2+2x+1. It involves graphing the equations to find the intersection points that bound the region, and then using the property that area is the difference of the areas under the curves to calculate it as the integral from 0 to 1 of -2x^2+2x dx.
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Areas between curves
- 1. Exploring areas between curvesTask Three
- 2. THE PROBLEMFind the area bounded by 𝑦=𝑥2+1 and 𝑦=−𝑥2+2𝑥+1.STEP 1: Graph the equations.
- 3. THE PROBLEMFind the area bounded by 𝑦=𝑥2+1 and 𝑦=−𝑥2+2𝑥+1.STEP 1: Graph the equations.
- 4. THE PROBLEMFind 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 PROBLEMFind 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 Purplewithin 0≤𝑥≤1
- 6. THE PROBLEMFind the area bounded by 𝑦=𝑥2+1 and 𝑦=−𝑥2+2𝑥+1.STEP 4: SOLVE!
- 7. THE SPECIAL PROPERTYFind the area bounded by 𝑦=𝑥2+1 and 𝑦=−𝑥2+2𝑥+1.𝒂𝒃𝒇𝒙±𝒈𝒙𝒅𝒙=𝒂𝒃𝒇𝒙𝒅𝒙±𝒂𝒃𝒈𝒙𝒅𝒙Area = (Area of Blue – Area of Purple)Area =𝑎𝑏𝑓𝑥𝑑𝑥−𝑎𝑏𝑔𝑥𝑑𝑥Area =𝑎𝑏𝑓𝑥±𝑔𝑥𝑑𝑥Area=𝑎𝑏((−𝑥2+2𝑥+1) − (𝑥2+1))𝑑𝑥Area= 𝑎𝑏(−2𝑥2+2𝑥)𝑑𝑥
- 8. THE SPECIAL PROPERTYFind the area bounded by 𝑦=𝑥2+1 and 𝑦=−𝑥2+2𝑥+1.*CHECK IF ANSWERS ARE CONSISTENT.Area= 𝑎𝑏(−2𝑥2+2𝑥)𝑑𝑥