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The area bounded by the curve y = x + sin x and its inverse function between the
ordinates x = 0 and x = 2π, is -
ordinates x = 0 and x = 2π, is -
- a)8π sq unit
- b)4π sq unit
- c)8 sq unit
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
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The area bounded by the curve y = x + sin x and its inverse function b...
To find the area bounded by the curve y = x * sin(x) and its inverse function between the ordinates x = 0 and x = 2, we need to find the difference between the two curves and integrate it over the given interval.
First, let's find the inverse function of y = x * sin(x).
To find the inverse, we swap x and y and solve for y:
x = y * sin(y)
y * sin(y) = x
sin(y) = x/y
Now, let's solve for y by taking the inverse sine of both sides:
y = arcsin(x/y)
Now, we can find the difference between the two functions:
f(x) = x * sin(x)
g(x) = arcsin(x/y)
The area bounded by the two curves between x = 0 and x = 2 is given by the integral of the difference function:
Area = ∫[0, 2] (f(x) - g(x)) dx
Area = ∫[0, 2] (x * sin(x) - arcsin(x/y)) dx
To evaluate this integral, we will need to use numerical methods or approximations since there is no elementary antiderivative for the difference function.
Therefore, the exact value of the area bounded by the curve y = x * sin(x) and its inverse function between the ordinates x = 0 and x = 2 cannot be determined analytically.
First, let's find the inverse function of y = x * sin(x).
To find the inverse, we swap x and y and solve for y:
x = y * sin(y)
y * sin(y) = x
sin(y) = x/y
Now, let's solve for y by taking the inverse sine of both sides:
y = arcsin(x/y)
Now, we can find the difference between the two functions:
f(x) = x * sin(x)
g(x) = arcsin(x/y)
The area bounded by the two curves between x = 0 and x = 2 is given by the integral of the difference function:
Area = ∫[0, 2] (f(x) - g(x)) dx
Area = ∫[0, 2] (x * sin(x) - arcsin(x/y)) dx
To evaluate this integral, we will need to use numerical methods or approximations since there is no elementary antiderivative for the difference function.
Therefore, the exact value of the area bounded by the curve y = x * sin(x) and its inverse function between the ordinates x = 0 and x = 2 cannot be determined analytically.
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The area bounded by the curve y = x + sin x and its inverse function between theordinates x = 0 and x = 2π, is -a)8π sq unitb)4π sq unitc)8 sq unitd)None of theseCorrect answer is option 'C'. Can you explain this answer?
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