Find the best tutors and institutes for Class 12 Tuition
Q3:
Area lying in the first quadrant and bounded by the circle $x^2 + y^2 = 4$ and the lines $x = 0$ and $x = 2$ is
Solution :
Step 1: Analyzing the Geometric Constraints
The given equation is $x^2 + y^2 = 4$, which represents a circle centered at the origin $(0, 0)$ with radius $r = \sqrt{4} = 2$ units [By the standard form of circle equation $(x-h)^2 + (y-k)^2 = r^2$].
The boundary conditions are:
- The circle: $y^2 = 4 - x^2 \implies y = \sqrt{4 - x^2}$ (taking the positive root as we are constrained to the first quadrant).
- The vertical lines: $x = 0$ (the $y$-axis) and $x = 2$ (the rightmost edge of the circle).
Step 2: Defining the Integral Setup
The area $A$ in the first quadrant bounded by the curve $f(x) = \sqrt{4 - x^2}$ from $x = 0$ to $x = 2$ is given by the definite integral:
$A = \int_{0}^{2} \sqrt{4 - x^2} \, dx$
Step 3: Evaluating the Integral
Using the standard integration formula $\int \sqrt{a^2 - x^2} \, dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1}\left(\frac{x}{a}\right) + C$:
Substitute $a = 2$:
$A = \left[ \frac{x}{2}\sqrt{4 - x^2} + \frac{4}{2} \sin^{-1}\left(\frac{x}{2}\right) \right]_{0}^{2}$
$A = \left[ \frac{x}{2}\sqrt{4 - x^2} + 2 \sin^{-1}\left(\frac{x}{2}\right) \right]_{0}^{2}$
Applying the Fundamental Theorem of Calculus:
$A = \left( \frac{2}{2}\sqrt{4 - 2^2} + 2 \sin^{-1}\left(\frac{2}{2}\right) \right) - \left( \frac{0}{2}\sqrt{4 - 0^2} + 2 \sin^{-1}\left(\frac{0}{2}\right) \right)$
$A = (1 \cdot 0 + 2 \cdot \sin^{-1}(1)) - (0 + 2 \cdot 0)$
$A = 2 \cdot \left(\frac{\pi}{2}\right) = \pi$
Step 4: Geometric Visualization
Final Solution: The area is $\pi$ square units.
More Questions from Class 12 Mathematics Application of Integrals EXERCISE 8.1
CBSE Solutions for Class 12 Mathematics Application of Integrals
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Application of Integrals
Top Tutor
Kalyan Nagar, Bangalore
12 yrs of
Exp
200
per hour
Online Classes
I have 14 years experience in hindi tuition , lots of student passed out in hindi with 90+. 9th - 12th all types of syllabus handled . Bangalore colleges like frank's, clarence , carmel, cmr, royal concorde , delhi public , charles and other schools. Online classes anywhere in india .
Top Tutor
Kharghar Sector 27, Mumbai
13 yrs of
Exp
6 students
1082 hrs of classes completed
650
per hour
Online Classes
Student's Home
Tutor's home
My approach to teaching Mathematics and Science is personalized and interactive. I focus on understanding each student's unique learning style and adapt my methods accordingly. My goal is to make Math and Science relatable and engaging, using real-life examples and practical applications to enhance understanding. I try to develop self-thinking on problems and encourage them to solve problems. **Key Skills:** - Strong grasp of Mathematics concepts across various levels - Ability to simplify complex topics for better understanding - Excellent communication and motivational skills - Patient and supportive, ensuring a comfortable learning environment - Regular progress assessments and constructive feedback **Services Offered:** - One-on-one personalized tutoring sessions - Homework and assignment assistance - Exam preparation and revision classes - Online and in-person tutoring options - Flexible scheduling to accommodate student needs **Achievements:** - Helped students improve their grades by an average of 25% - Successfully prepared students for competitive exams with success. Received positive feedback and high ratings from students and parents
He teaches from basics.Built up lot of confidence.Took lot of paper practice.Explains by giving practical life examples. Must go for it.
Top Tutor
HSR Layout, Bangalore
20 yrs of
Exp
30 students
1728 hrs of classes completed
700
per hour
Online Classes
Tutor's home
With a distinguished Doctorate in Chemistry, I bring 28 years of expertise, seamlessly integrating profound knowledge and unparalleled teaching prowess. As a Chemistry Teacher for CBSE and IBDP Chemistry Faculty, I've significantly contributed to intellectual growth in prestigious institutions like Kendriya Vidyalaya, Ecole Mondiale World School, and Singapore International School etc. Recognized as a trailblazer, I've served as an Assistant Examiner and Moderator for Internal Assessment in IBDP Chemistry for six years, upholding the highest evaluation standards and showcasing a deep understanding of curriculum intricacies. My extensive experiences beyond conventional teaching establish me as a luminary in Chemistry education, leaving an enduring legacy in the annals of education. I remain dedicated to inspiring future generations to achieve unparalleled heights in scientific inquiry and knowledge.
A very knowledgeable and caring teacher Patient and passionate teaching. All these qualities helped me succeed and score well in my board exams.
Top Tutor
Bhelagaon Rd Bhela, Bela Bhela
2 yrs of
Exp
2 students
33 hrs of classes completed
500
per hour
Online Classes
I am online tutor providing tuition for students and i have good experience and knowledge of teaching students and knowledge of subjects. I will help you in getting best marks in examination and in understanding the subjects
Super Tutor
H C Sarani, Kolkata
18 yrs of
Exp
16 students
2047 hrs of classes completed
600
per hour
Online Classes
Tutor's home
Great
Sir is a dedicated and knowledgeable teacher who explains complex topics with clarity and patience. His classes are well-structured, engaging, and focused on practical understanding as well as theory. Overall, a highly effective and professional educator.
Other Subjects in CBSE - Class 12
Download free CBSE - Class 12 Mathematics Application of Integrals EXERCISE 8.1 worksheets
Download Now
