Exploring areas between curves<br />Task Three<br />
THE PROBLEM<br />Find the area bounded by 𝑦=π‘₯2+1Β and 𝑦=βˆ’π‘₯2+2π‘₯+1.<br />STEP 1: Graph the equations.<br />Β <br />
THE PROBLEM<br />Find the area bounded by 𝑦=π‘₯2+1Β and 𝑦=βˆ’π‘₯2+2π‘₯+1.<br />STEP 1: Graph the equations.<br />Β <br />
THE PROBLEM<br />Find the area bounded by 𝑦=π‘₯2+1Β and 𝑦=βˆ’π‘₯2+2π‘₯+1.<br />STEP 2: Find the intersections to determine the x va...
THE PROBLEM<br />Find the area bounded by 𝑦=π‘₯2+1Β and 𝑦=βˆ’π‘₯2+2π‘₯+1.<br />STEP 3: Use logic to determine the best way to get t...
THE PROBLEM<br />Find the area bounded by 𝑦=π‘₯2+1Β and 𝑦=βˆ’π‘₯2+2π‘₯+1.<br />STEP 4: SOLVE!<br />Β <br />
THE SPECIAL PROPERTY<br />Find the area bounded by 𝑦=π‘₯2+1Β and 𝑦=βˆ’π‘₯2+2π‘₯+1.<br />π’‚π’ƒπ’‡π’™Β±π’ˆπ’™π’…π’™=π’‚π’ƒπ’‡π’™π’…π’™Β±π’‚π’ƒπ’ˆπ’™π’…π’™<br />Area = (Area o...
THE SPECIAL PROPERTY<br />Find the area bounded by 𝑦=π‘₯2+1Β and 𝑦=βˆ’π‘₯2+2π‘₯+1.<br />*CHECK IF ANSWERS ARE CONSISTENT.<br />Area...
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Areas between curves

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Areas between curves

  1. 1. Exploring areas between curves<br />Task Three<br />
  2. 2. THE PROBLEM<br />Find the area bounded by 𝑦=π‘₯2+1Β and 𝑦=βˆ’π‘₯2+2π‘₯+1.<br />STEP 1: Graph the equations.<br />Β <br />
  3. 3. THE PROBLEM<br />Find the area bounded by 𝑦=π‘₯2+1Β and 𝑦=βˆ’π‘₯2+2π‘₯+1.<br />STEP 1: Graph the equations.<br />Β <br />
  4. 4. THE PROBLEM<br />Find the area bounded by 𝑦=π‘₯2+1Β and 𝑦=βˆ’π‘₯2+2π‘₯+1.<br />STEP 2: Find the intersections to determine the x values which bound the region of the unknown area.<br />Β <br />(1, 2)<br />(0, 1)<br />
  5. 5. THE PROBLEM<br />Find the area bounded by 𝑦=π‘₯2+1Β and 𝑦=βˆ’π‘₯2+2π‘₯+1.<br />STEP 3: Use logic to determine the best way to get the area.<br />Β <br />(1, 2)<br />(0, 1)<br />Area of Yellow = <br />Area of Blue – Area of Purple<br />within 0≀π‘₯≀1<br />Β <br />
  6. 6. THE PROBLEM<br />Find the area bounded by 𝑦=π‘₯2+1Β and 𝑦=βˆ’π‘₯2+2π‘₯+1.<br />STEP 4: SOLVE!<br />Β <br />
  7. 7. THE SPECIAL PROPERTY<br />Find the area bounded by 𝑦=π‘₯2+1Β and 𝑦=βˆ’π‘₯2+2π‘₯+1.<br />π’‚π’ƒπ’‡π’™Β±π’ˆπ’™π’…π’™=π’‚π’ƒπ’‡π’™π’…π’™Β±π’‚π’ƒπ’ˆπ’™π’…π’™<br />Area = (Area of Blue – Area of Purple)<br />Area =π‘Žπ‘π‘“π‘₯𝑑π‘₯βˆ’π‘Žπ‘π‘”π‘₯𝑑π‘₯<br />Area =π‘Žπ‘π‘“π‘₯±𝑔π‘₯𝑑π‘₯<br />Area=π‘Žπ‘((βˆ’π‘₯2+2π‘₯+1)Β βˆ’Β (π‘₯2+1))𝑑π‘₯<br />Area=Β π‘Žπ‘(βˆ’2π‘₯2+2π‘₯)𝑑π‘₯<br />Β <br />
  8. 8. THE SPECIAL PROPERTY<br />Find the area bounded by 𝑦=π‘₯2+1Β and 𝑦=βˆ’π‘₯2+2π‘₯+1.<br />*CHECK IF ANSWERS ARE CONSISTENT.<br />Area=Β π‘Žπ‘(βˆ’2π‘₯2+2π‘₯)𝑑π‘₯<br />Β <br />

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