Porisma

A hint to get started

Reuse a substitution result in a circle-area problem

(i) Using x=sin⁡θx=\sin\theta, show that ∫01/21−x2 dx=π/12+3/8\int_0^{1/2}\sqrt{1-x^2}\,dx=\pi/12+\sqrt3/8. (ii) The circle x2+(y−1)2=1x^2+(y-1)^2=1 and the line y=1−3xy=1-\sqrt3x intersect at A=(1/2,1−3/2)A=(1/2,1-\sqrt3/2). Using part (i), find the exact area of the shaded region bounded by the circle, line and x-axis.

[2018/AJC/Promo/8]

CJC · Lecture · Example 20

Both copies already print the full substitution, branch-selection box and area derivation. Teacher 25 highlights exact area; Teacher 26 circles the triangular term and labels it area of triangle. These emphases are integrated into the exact-answer and decomposition steps.

The source supplies A and its coordinates, so use that join point to split the two different upper boundaries.

Read the worked solution