Question

Find the area bounded by the curve y= 4x - x2 , the x axis and the ordinates x=1 and x=3

Answer 1

Here, shaded region represents the area bounded by y=4x−x2 and y=0

Area of the shaded region:

Let A be the area of the shaded region.

Then

A=∫ba[f(x)−g(x)]dx

Here, f(x) is the top curve and g(x) is the bottom curve.

a and b are the limits as

a (Lower limit) = x coordinate of extreme left intersection point of area to be found.

b (Upper limit) = x coordinate of extreme right intersection point of area to be found.

So, f(x)=y=4x−x2

g(x)=y=0

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 of the given curves.

∴ We will find the intersection points of the given curve and line.

Put the value of y from equation (ii) into equation (i)

4x−x2=0

x(4−x)=0

x=0,x=4

Put these values in equation (i)

y=0,y=0

Thus the points of intersection are O(0,0) and A(4,0)

∴,a=0,b=4

Area between curves:

A=∫ba[f(x)−g(x)]dx=∫40[(4x−x2)−0]dx

=[4(x22−x33)]=[32−643]−0

=[96−643]

A=323First, we will find the area of the region bounded by the curve and x-axis.

At x-axis,

So, we find the area between:

… (i)


To determine the area of the shaded region between curve and x-axis, we need to sketch this curve on the graph.

Answer 2

Answer: 22/3

Step-by-step explanation: