what is the area of the region
Answers
Answer:
First we will find the area of the region bounded by the curves:
y=x2 ... (i)
and y=x ... (ii)
To determine the shaded area between these two curves, we need to sketch these curves on a graph. enter image description here
Now, we will find the area of the shaded region from O to A.
Area of Shaded Region Between Two Curves :
A=∫ba[f(x)−g(x)]dx
Where, f(x) is the top curve
g(x) is the bottom curve
a (Lower limit) = x coordinate of extreme left intersection point of the area to be found.
b (Upper limit) = x coordinate of extreme right intersection point of the area to be found.
So, f(x)=y=x
g(x)=y=x2
We need to find the limits, a and b.
How to find the limits ?
Since limits, a and b, are the x coordinates of the intersection points, So, we will find the intersection points of the given curves.
Put the value of y from equation (ii) into equation (i)
x=x2
x2−x=0
x(x−1)=0
x=0,x=1
Put these values in equation (ii)
y=0,y=1
Thus, the points of intersection are O(0,0) and A(1,1)
∴a=0,b=1
Area between Curves :
The are will be, A=∫ba[f(x)−g(x)]dx
A=∫10(x−x2)dx
A=∫10xdx−∫10x2dx
=(x22)10−(x33)10
On putting limits,
=(12−0)−(13−0)
=12−13
A=16
(II) Now, we will find the shaded area bounded by the curves:
y=x2+1 ... (iii)
y=2 ... (iv)
Curves on Graph : enter image description here
We will find the area of the shaded region from A to B
Here, f(x)=y=2
g(x)=y=x2+1
Finding the limits using intersection points :
Put the value of y in equation (iii)
2=x2+1
x2=1
x=−1,x=1
Put these values in equation (iii)
y=2,y=2
Thus, the points of intersection are A(−1,2) and B(1,2)
∴a=−1,b=1
Area between Curves :
The area will be,
A=∫ba[f(x)−g(x)]dx
A=∫1−1[2−(x2+1)]dx
A=∫1−11dx−∫1−1x2dx
=(x)1−1−(x33)1−1
On putting limits, we get,
=(1+1)−(13+13)
=2−23
A=43