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Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Area Under Curves

Olympiad workbook · IOQM · Area Under Curves

Let R denote the circular region in the x y -plane bounded by the...

Let RR denote the circular region in the xyx y-plane bounded by the circle x2+y2=36x^2+y^2=36. The lines x=4andx=4 andy=3dividedivideRintofourregionsinto four regionsR_i, i=1,2,3,4.If∣Ri∣, 4 . If \left|R_i\right| denotes the area of the region RiandifR_i and if\left|\mathrm{R}_1\right|>\left|\mathrm{R}_2\right|>\left|\mathrm{R}_3\right|>\left|\mathrm{R}_4\right|,determine∣R1∣−∣R2∣−∣R3∣+∣R4∣, determine \left|\mathrm{R}_1\right|-\left|\mathrm{R}_2\right|-\left|\mathrm{R}_3\right|+\left|\mathrm{R}_4\right|. [Here ∣Ω∣|\Omega| denotes the area of the region Ω\Omega in the plane.]Let R denote the circular region in the x y -plane bounded by the circle x^2+y^2=36 . The lines x=4 and y=3 divide R

Correct answer

48

Step-by-step solution

Let R denote the circular region in the x y -plane bounded by the circle x^2+y^2=36 . The lines x=4 and y=3 divide R
R1=∫−25336−y2dy=12y36−y2+12(36)sin⁡−1y6R_1=\int_{-2 \sqrt{5}}^3 \sqrt{36-y^2} d y=\frac{1}{2} y \sqrt{36-y^2}+\frac{1}{2}(36) \sin ^{-1} \frac{y}{6} Similarly for using integration. we can find R1+R4=12π(6)2+2(3)(4)∴ Remaining circle area =R2+R3=12π(6)2−2(3)(4)\begin{gathered} \mathrm{R}_1+\mathrm{R}_4=\frac{1}{2} \pi(6)^2+2(3)(4) \\ \therefore \text { Remaining circle area }=\mathrm{R}_2+\mathrm{R}_3=\frac{1}{2} \pi(6)^2-2(3)(4) \end{gathered} Hence, R_1+R_4-\left(R_2+R_3\right)=24+24=48
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