
Find the area bounded by and its inverse between and .
A) 2
B) 4
C) 6
D) 8
Answer
482.4k+ views
Hint:
Here, we will use the concept of integration to find the area bounded by the curves. First we will find the point of inflection and then integrate the curves to find the area. Integration is the process of adding small areas to find the whole area.
Formula Used: Area between two curves which intersect each other at is given by
Complete step by step solution:
We are given with a function .
Replacing the function with , we get
Here, .
Differentiating with respect to , we get
,
Again differentiating with respect to , we get
We know that at all integers of .
Therefore, is the point of inflection for .
So, we have , and .
Also , and .
Using these we can draw a graph as shown below:
Area between two curves which intersect each other at is given by .
Now,
Area bounded by and its inverse
Inverse function is symmetrical with respect to .
Thus by using the property of symmetry, we get
Area bounded by and its inverse
Substituting the function , we get
Area bounded by and its inverse
Subtracting the terms in the integrand, we get
Area bounded by and its inverse
Integrating the function, we get
Area bounded by and its inverse
Substituting the upper limit and lower limit, we get
Area bounded by and its inverse
Adding and multiplying, we get
Area bounded by and its inverse
Area bounded by and its inverse
Adding the terms, we get
Area bounded by and its inverse sq. units
Therefore, the area bounded by and its inverse between and is 8 sq. units.
Note:
We need to keep in mind that, while doing integration, we should notice from the graph the curve at the bottom should be subtracted from the curve at the top. We should also know that the inflection point is a point where a function changes its sign and its direction. We should remember that the function has to be differentiated twice to find the inflection points. The limits of the integral become when both the curves are symmetrical to each other.
Here, we will use the concept of integration to find the area bounded by the curves. First we will find the point of inflection and then integrate the curves to find the area. Integration is the process of adding small areas to find the whole area.
Formula Used: Area between two curves which intersect each other at
Complete step by step solution:
We are given with a function
Replacing the function
Here,
Differentiating with respect to
Again differentiating with respect to
We know that
Therefore,
So, we have
Also
Using these we can draw a graph as shown below:

Area between two curves which intersect each other at
Now,
Area bounded by
Inverse function is symmetrical with respect to
Thus by using the property of symmetry, we get
Substituting the function
Subtracting the terms in the integrand, we get
Integrating the function, we get
Substituting the upper limit and lower limit, we get
Adding and multiplying, we get
Adding the terms, we get
Therefore, the area bounded by
Note:
We need to keep in mind that, while doing integration, we should notice from the graph the curve at the bottom should be subtracted from the curve at the top. We should also know that the inflection point is a point where a function changes its sign and its direction. We should remember that the function has to be differentiated twice to find the inflection points. The limits of the integral become when both the curves are symmetrical to each other.
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