<P> For each two - dimensional shape below, the area and the centroid coordinates (x _̄, y _̄) (\ displaystyle ((\ bar (x)), (\ bar (y)))) are given: </P> <Table> <Tr> <Th> Shape </Th> <Th> Figure </Th> <Th> x _̄ (\ displaystyle (\ bar (x))) </Th> <Th> y _̄ (\ displaystyle (\ bar (y))) </Th> <Th> Area </Th> </Tr> <Tr> <Td> Right - triangular area </Td> <Td> </Td> <Td> b 3 (\ displaystyle (\ frac (b) (3))) </Td> <Td> h 3 (\ displaystyle (\ frac (h) (3))) </Td> <Td> b h 2 (\ displaystyle (\ frac (bh) (2))) </Td> </Tr> <Tr> <Td> Quarter - circular area </Td> <Td> </Td> <Td> 4 r 3 π (\ displaystyle (\ frac (4r) (3 \ pi))) </Td> <Td> 4 r 3 π (\ displaystyle (\ frac (4r) (3 \ pi))) </Td> <Td> π r 2 4 (\ displaystyle (\ frac (\ pi r ^ (2)) (4))) </Td> </Tr> <Tr> <Td> Semicircular area </Td> <Td> </Td> <Td> 0 </Td> <Td> 4 r 3 π (\ displaystyle (\ frac (4r) (3 \ pi))) </Td> <Td> π r 2 2 (\ displaystyle (\ frac (\ pi r ^ (2)) (2))) </Td> </Tr> <Tr> <Td> Quarter - elliptical area </Td> <Td> </Td> <Td> 4 a 3 π (\ displaystyle (\ frac (4a) (3 \ pi))) </Td> <Td> 4 b 3 π (\ displaystyle (\ frac (4b) (3 \ pi))) </Td> <Td> π a b 4 (\ displaystyle (\ frac (\ pi ab) (4))) </Td> </Tr> <Tr> <Td> Semielliptical area </Td> <Td> </Td> <Td> 0 (\ displaystyle \, \! 0) </Td> <Td> 4 b 3 π (\ displaystyle (\ frac (4b) (3 \ pi))) </Td> <Td> π a b 2 (\ displaystyle (\ frac (\ pi ab) (2))) </Td> </Tr> <Tr> <Td> Semiparabolic area <P> The area between the curve y = h b 2 x 2 (\ displaystyle y = (\ frac (h) (b ^ (2))) x ^ (2)) and the y (\ displaystyle \, \! y) axis, from y = 0 (\ displaystyle \, \! y = 0) to y = h (\ displaystyle \, \! y = h) </P> </Td> <Td> </Td> <Td> 3 b 8 (\ displaystyle (\ frac (3b) (8))) </Td> <Td> 3 h 5 (\ displaystyle (\ frac (3h) (5))) </Td> <Td> 2 b h 3 (\ displaystyle (\ frac (2bh) (3))) </Td> </Tr> <Tr> <Td> Parabolic area </Td> <Td> The area between the curve y = h b 2 x 2 (\ displaystyle \, \! y = (\ frac (h) (b ^ (2))) x ^ (2)) and the line y = h (\ displaystyle \, \! y = h) </Td> <Td> 0 (\ displaystyle \, \! 0) </Td> <Td> 3 h 5 (\ displaystyle (\ frac (3h) (5))) </Td> <Td> 4 b h 3 (\ displaystyle (\ frac (4bh) (3))) </Td> </Tr> <Tr> <Td> Parabolic spandrel </Td> <Td> The area between the curve y = h b 2 x 2 (\ displaystyle \, \! y = (\ frac (h) (b ^ (2))) x ^ (2)) and the x (\ displaystyle \, \! x) axis, from x = 0 (\ displaystyle \, \! x = 0) to x = b (\ displaystyle \, \! x = b) </Td> <Td> 3 b 4 (\ displaystyle (\ frac (3b) (4))) </Td> <Td> 3 h 10 (\ displaystyle (\ frac (3h) (10))) </Td> <Td> b h 3 (\ displaystyle (\ frac (bh) (3))) </Td> </Tr> <Tr> <Td> General spandrel </Td> <Td> The area between the curve y = h b n x n (\ displaystyle y = (\ frac (h) (b ^ (n))) x ^ (n)) and the x (\ displaystyle \, \! x) axis, from x = 0 (\ displaystyle \, \! x = 0) to x = b (\ displaystyle \, \! x = b) </Td> <Td> n + 1 n + 2 b (\ displaystyle (\ frac (n + 1) (n + 2)) b) </Td> <Td> n + 1 4 n + 2 h (\ displaystyle (\ frac (n + 1) (4n + 2)) h) </Td> <Td> b h n + 1 (\ displaystyle (\ frac (bh) (n + 1))) </Td> </Tr> <Tr> <Td> Circular sector </Td> <Td> </Td> <Td> 2 r sin ⁡ (α) 3 α (\ displaystyle (\ frac (2r \ sin (\ alpha)) (3 \ alpha))) </Td> <Td> 0 (\ displaystyle \, \! 0) </Td> <Td> α r 2 (\ displaystyle \, \! \ alpha r ^ (2)) </Td> </Tr> <Tr> <Td> Circular segment </Td> <Td> </Td> <Td> 4 r sin 3 ⁡ (α) 3 (2 α − sin ⁡ (2 α)) (\ displaystyle (\ frac (4r \ sin ^ (3) (\ alpha)) (3 (2 \ alpha - \ sin (2 \ alpha))))) </Td> <Td> 0 (\ displaystyle \, \! 0) </Td> <Td> r 2 2 (2 α − sin ⁡ (2 α)) (\ displaystyle (\ frac (r ^ (2)) (2)) (2 \ alpha - \ sin (2 \ alpha))) </Td> </Tr> <Tr> <Td> Quarter - circular arc </Td> <Td> The points on the circle x 2 + y 2 = r 2 (\ displaystyle \, \! x ^ (2) + y ^ (2) = r ^ (2)) and in the first quadrant </Td> <Td> 2 r π (\ displaystyle (\ frac (2r) (\ pi))) </Td> <Td> 2 r π (\ displaystyle (\ frac (2r) (\ pi))) </Td> <Td> L = π r 2 (\ displaystyle L = (\ frac (\ pi r) (2))) </Td> </Tr> <Tr> <Td> Semicircular arc </Td> <Td> The points on the circle x 2 + y 2 = r 2 (\ displaystyle \, \! x ^ (2) + y ^ (2) = r ^ (2)) and above the x (\ displaystyle \, \! x) axis </Td> <Td> 0 (\ displaystyle \, \! 0) </Td> <Td> 2 r π (\ displaystyle (\ frac (2r) (\ pi))) </Td> <Td> L = π r (\ displaystyle L = \, \! \ pi r) </Td> </Tr> <Tr> <Td> Arc of circle </Td> <Td> The points on the curve (in polar coordinates) r = ρ (\ displaystyle \, \! r = \ rho), from θ = − α (\ displaystyle \, \! \ theta = - \ alpha) to θ = α (\ displaystyle \, \! \ theta = \ alpha) </Td> <Td> ρ sin ⁡ (α) α (\ displaystyle (\ frac (\ rho \ sin (\ alpha)) (\ alpha))) </Td> <Td> 0 (\ displaystyle \, \! 0) </Td> <Td> L = 2 α ρ (\ displaystyle L = \, \! 2 \ alpha \ rho) </Td> </Tr> </Table> <Tr> <Th> Shape </Th> <Th> Figure </Th> <Th> x _̄ (\ displaystyle (\ bar (x))) </Th> <Th> y _̄ (\ displaystyle (\ bar (y))) </Th> <Th> Area </Th> </Tr> <Tr> <Td> Right - triangular area </Td> <Td> </Td> <Td> b 3 (\ displaystyle (\ frac (b) (3))) </Td> <Td> h 3 (\ displaystyle (\ frac (h) (3))) </Td> <Td> b h 2 (\ displaystyle (\ frac (bh) (2))) </Td> </Tr>

The center of mass of quarter circle given by