Q. 305.0( 2 Votes )

Using integration find the area of the triangle formed by positive x-axis and tangent and normal to the circle x2 + y2 = 4 at (1, √3).
OR

Evaluate: as a limit of a sum.  [CBSE 2015]

Answer :

As first we need to trace the area to be determined.As x2+y2 = 4 represents a circle whose centre is at (0,0) and radius = 2 cm. The rough sketch is shown below:


As normal at (1,√3) passes through origin too because in a circle a normal always passes through centre of the circle.


Equation of normal is y = √3 x


Similarly equation of tangent can be written using one point and slope form.


As, x2 + y2 = 4


Differentiating w.r.t x:






We need to determine the area enclosed i.e. area(region ABC).


Area(region ABC) = area(region ABD) + area(region BDC)


Area = area under curve y = √3x + area under


Required area =


Area =


Area =


Area =


Required area =


OR


We know that a definite integral can be evaluated as a limit of a sum as-



Where h =


As we have to find:


Let I = and on comparing I with the formula we can say that a = 1 and b = 3.


I =


I = 2


I = 2


I = 2


Each bracket contains ‘n’ terms.


I = 2


I = 2


I = 2


Using formula for sum of first n natural numbers; sum of squares of first n natural numbers and sun of n terms of a GP we get:


I = 2


I = 2


As h =


I = 2


I = 2


Using algebra of limits:


I = 2


Use the formula:


I = 2


I =

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Fundamental Integration FormulaFundamental Integration FormulaFundamental Integration Formula59 mins
Interactive Quiz on Integration by SubstitutionInteractive Quiz on Integration by SubstitutionInteractive Quiz on Integration by Substitution47 mins
Lecture on Integration by partsLecture on Integration by partsLecture on Integration by parts55 mins
Lecture on some forms of integrationLecture on some forms of integrationLecture on some forms of integration54 mins
Lecture on integration by partial fractionsLecture on integration by partial fractionsLecture on integration by partial fractions62 mins
Interactive Quiz on Integration by PartsInteractive Quiz on Integration by PartsInteractive Quiz on Integration by Parts56 mins
Integration by SubstitutionIntegration by SubstitutionIntegration by Substitution56 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses
RELATED QUESTIONS :

is equal to

Mathematics - Exemplar

Evaluate the following:

Mathematics - Exemplar

is equal to

Mathematics - Exemplar

Evaluate the following:

Mathematics - Exemplar