1. Section 7.1 Day 1-2 Area of a Region Between Two Curves
AP Calculus AB

2. Learning Targets
Find the area between two curves using a vertical cross section
Find the area between two intersecting curves
Find the area between two curves using a horizontal cross section

3. Area Between Two Curves Formula
𝐴= 𝑎 𝑏 𝑓 𝑥 −𝑔 𝑥 𝑑𝑥

4. Example 1
Find the area of the region bounded by the graphs of 𝑦= 𝑥 2 +2, 𝑦=−𝑥, 𝑥=0, and 𝑥=1
0 1 𝑥 2 +2 −(−𝑥) 𝑑𝑥
0 1 𝑥 2 +𝑥+2 𝑑𝑥
1 3 𝑥 𝑥 2 +2𝑥 from 1 to 0
2.833 𝑜𝑟 17 6

5. Example 2: Intersecting Curves
Find the area of the region bounded by the graphs of 𝑓 𝑥 =2− 𝑥 2 and 𝑔 𝑥 =𝑥
Find Intersections: 2− 𝑥 2 =𝑥
𝑥 2 +𝑥−2=0
𝑥+2 𝑥−1 =0
𝑥=−2, 𝑥=1
−2 1 2− 𝑥 2 −𝑥 𝑑𝑥=4.5

6. Example 3: More than 2 Intersections
Find the area of the region between the graphs of 𝑓 𝑥 =3 𝑥 3 − 𝑥 2 −10𝑥 and 𝑔 𝑥 =− 𝑥 2 +2𝑥
1. Find intersections: 3 𝑥 3 − 𝑥 2 −10𝑥=− 𝑥 2 +2𝑥
2. 3 𝑥 3 −12𝑥=0, 3𝑥 𝑥 2 −4 =0, 𝑥=0, ±2
3. −2 0 𝑓 𝑥 −𝑔 𝑥 𝑑𝑥 =12
𝑔 𝑥 −𝑓 𝑥 𝑑𝑥 =12
5. Add together for a total of 24

7. Example 4: More than 2 Intersections
Find the area of the region between the graphs of 𝑓 𝑥 = 𝑥 3 −6𝑥+2 and 𝑔 𝑥 =2𝑥
1. Use Calculator to find intersections
2. 𝑥=2.694, 𝑥=0.252, 𝑥=−2.946
3. − 𝑓 𝑥 −𝑔 𝑥 𝑑𝑥 =
𝑔 𝑥 −𝑓 𝑥 𝑑𝑥 =
5. Add together:

8. Example 5: Sub Regions
Find the area of the region bounded by the graphs of 𝑥=3− 𝑦 2 and 𝑥=𝑦+1
1. Solve in terms of y: 𝑦=𝑥−1, 𝑦=± 3−𝑥
2. Split into 2 parts at 𝑥=2
3. −1 2 𝑥−1 −(− 3−𝑥 ) 𝑑𝑥=19/6
−𝑥 𝑑𝑥=1.3333
5. Add together: 4.500

9. Example 6: Sub Regions
Find the area of the region R in the first quadrant that is bounded above by 𝑦= 𝑥 and below by the x-axis and the line 𝑦=𝑥−2
1. Split into 2 regions at 𝑥=2
𝑥 𝑑𝑥=
𝑥 − 𝑥−2 𝑑𝑥 =
4. Add together: 3.333

10. Example 7: Horizontal Cross Sections
Find the area of the region bounded by the graphs of 𝑥=3− 𝑦 2 and 𝑥=𝑦+1
1. Find bounds: 𝑦=−2, 𝑦=1
2. −2 1 3− 𝑦 2 − 𝑦+1 𝑑𝑦 =4.5

11. Example 8: Horizontal Cross Sections
Find the area enclosed by the graphs of 𝑦= 𝑥 3 and 𝑥= 𝑦 2 −2
1. Solve for x: 𝑥= 𝑦 2 −2, 𝑥= 𝑦 1 3
2. Use calculator to find intersections:
3. 𝑥=−1, 1.793
4. − 𝑦 − 𝑦 2 −2 𝑑𝑦
5. =4.215