The area of the region bounded by the curve y=x2−4x+3 and y=3−x is ....
Explanation
To calculate the area of the region bounded by both curves, first find the intersection points between the parabola and the line.
Finding Intersection Points
The intersection points are obtained by equating both equations
x2−4x+3=3−x
x2−3x=0
x(x−3)=0
Therefore, x1=0 or x2=3
Determining Upper and Lower Curves
In the interval between x=0 and x=3, the line y=3−x is above the parabola y=x2−4x+3
Graph Visualization
Graph of y=x2−4x+3 and y=3−x
Visualization of both curves forming a bounded region with intersection points.
Calculating the Area
The area of the region is calculated using integration
L=∫x1x2(yupper−ylower)dx
=∫03[(3−x)−(x2−4x+3)]dx
=∫03(−x2+3x)dx
=[−31x3+23x2]03
=(−31(3)3+23(3)2)−(−31(0)3+23(0)2)
=−9+227
=29
Therefore, the area of the region between the two curves is 29 square units.