# 707 Centroid of quarter ellipse by integration

**Problem 707**

Determine the centroid of the quadrant of the ellipse shown in Fig. P-707. The equation of the ellipse is $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$.

**Problem 707**

Determine the centroid of the quadrant of the ellipse shown in Fig. P-707. The equation of the ellipse is $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$.

**Problem 706**

Determine the centroid of the quarter circle shown in Fig. P-706 whose radius is r.

**Problem 705**

Determine the centroid of the shaded area shown in Fig. P-705, which is bounded by the x-axis, the line x = a and the parabola y^{2} = kx.

$W \, \bar{x} = \Sigma wx$

$W \, \bar{y} = \Sigma wy$

$A \, \bar{x} = \Sigma ax$

$A \, \bar{y} = \Sigma ay$

$L \, \bar{x} = \Sigma lx$

$L \, \bar{y} = \Sigma ly$

$W \, \bar{x} = \Sigma wx$

$W \, \bar{y} = \Sigma wy$

$W \, \bar{z} = \Sigma wz$

$V \, \bar{x} = \Sigma vx$

$V \, \bar{y} = \Sigma vy$

$V \, \bar{z} = \Sigma vz$

**Problem**

From the figure shown below, DE is the diameter of circle A and BC is the radius of circle B. If DE = 60 cm and AC = 10 cm, find the area of the shaded region.

Sequence is a succession of numbers formed according to some fixed rule. Example is

$1,~ 8,~ 27,~ 64,~ 125,~ ...$

which is a sequence so that the n^{th} term is given by n^{3}.

Series is the indicated sum of a sequence of numbers. Thus,

$a_1 + a_2 + a_3 + ... + a_n + ...$

is the series corresponding to the sequence $a_1,~ a_2,~ a_3,~ ... ,~a_n,~ ...$

**Finite and Infinite Series**

Problem

The sum of the digits of a three-place number is 19. If the tens and units digits are interchanged the number is diminished by 27, and if the hundreds and tens digits are interchanged the number is increased by 180. What is the number?

**Problem**

The sum of the parent’s ages is twice the sum of their children’s ages. Five years ago, the sum of the parent’s ages is four times the sum of their children’s ages. In fifteen years, the sum of the parent’s ages will be equal to the sum of their children’s ages. How many children were in the family?

**Problem**

Alfred is four times as old as his nephew Franco. 5 years ago, the sum of their ages is equal to the present age of Alfred. How old is each?

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