Equations and Graphs ⑤ Area of Figures Enclosed by Graphs
Find the areas of triangles and trapezoids enclosed by the graphs of multiple lines. Find the coordinates of the intersections first, then use the area formula.
Example ①
Find the area of the triangle enclosed by y = x + 2, y = −2x + 8, and the x-axis.
Answer:
Intersection of y=x+2 and the x-axis: y=0→x=−2. Point (−2, 0).
Intersection of y=−2x+8 and the x-axis: y=0→x=4. Point (4, 0).
Intersection of the two lines: x+2=−2x+8→x=2,y=4. Point (2, 4).
Base = 4−(−2) = 6, height = 4. Area = (1/2)×6×4 = 12
Find the area of the triangle enclosed by y = x + 2, y = −2x + 8, and the x-axis.
Answer:
Intersection of y=x+2 and the x-axis: y=0→x=−2. Point (−2, 0).
Intersection of y=−2x+8 and the x-axis: y=0→x=4. Point (4, 0).
Intersection of the two lines: x+2=−2x+8→x=2,y=4. Point (2, 4).
Base = 4−(−2) = 6, height = 4. Area = (1/2)×6×4 = 12
Key Points
- First, find the coordinates of all intersections
- Identify the base and height (the difference between the two points on the x-axis = the base)
- Divide complex shapes into parts and calculate them separately
Practice Problems
- Find the area of the triangle enclosed by y = 2x, y = −x + 6, and the x-axis.
Answers and Explanations
- Answer: 12. The three vertices are (0,0), (6,0), and (2,4) (the intersection of y=2x and y=−x+6). Base=6, height=4. Area=(1/2)×6×4=12.