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Area Calculation of a Quarter Circle with Triangle Subtraction

This analysis explores the area function G(x) defined for the interval 0 ≤ x ≤ 3, where G(x) represents the shaded green area in the left diagram. The function is derived from the area of a quarter circle with a radius of 3, minus the area of a yellow shaded region R in the right diagram. The area of R is determined by subtracting the area of a right triangle from a sector of angle θ. Moreover, for the range -3 < x < 0, G(x) corresponds to the area of a symmetric shaded pink region, which is equivalent to G(-x) computed on the previous slide.

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Area Calculation of a Quarter Circle with Triangle Subtraction

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  1. Example 2 Find Solution For 0  x  3, G(x) is the shaded green area in the left diagram. Hence G(x) is the area of the quarter circle of radius 3 minus the area of the yellow shaded region R in the right diagram above. Observe that Hence the area of R is the area of the sector of angle  minus the area of the right triangle OPQ: Then G(0) = 0 and for 0<x 3: 

  2. For -3x<0, G(x) is related to the area of the shaded pink region above which by symmetry is the same as the area of the shaded green region G(-x) which we computed on the first slide. Observe that G(x) is given by different formulas for -3x<0 and for 0<x 3.

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