Sketch the region bounded by the curves `y=sqrt(5-x^2)` and y=|x-1| and find its area using integration.
Consider the given equation
This equation represents a semicircle with centre at
the origin and radius = sqrt5 units
Given that the region is bounded by the above
semicircle and the line y = |x-1|
Let us find the point of intersection of the
given curve meets the line y= |x - 1|
Squaring both the sides, we have,
When x = -1,y = 2
When x = 2,y = 1
Consider the following figure.
Thus the intersection points are ( -1,2) and (2,1)
Consider the following sketch of the bounded region.
Required Area, A= `int_(-1)^2(y_2-y_1)dx`
`=5/2 sin^-1 (1/sqrt5)+5/2 sin^-1 (2/sqrt5)-1/2`
Required area= `[5/2 sin^-1 (1/sqrt5)+5/2 sin^-1 (2/sqrt5)-1/2 ] sq.units`
Video Tutorials For All Subjects
- Area of the Region Bounded by a Curve and a Line