Question

problem 3: centroid
sketch the region bounded by the curves. find the exact coordinates of the centroid. show all your steps clearly.
y = 2√x, y = 0, x = 4
problem 4: hydrostatic force
a trough is filled with a liquid of density 875 kg/m³

Explanation:

Step1: Calculate the area \(A\) of the region

The region is bounded by \(y = 2\sqrt{x}\), \(y = 0\), and \(x = 4\). The area \(A\) is given by the integral \(\int_{a}^{b}f(x)dx\), where \(a = 0\), \(b=4\) and \(f(x)=2\sqrt{x}\).

$$A=\int_{0}^{4}2\sqrt{x}dx=2\int_{0}^{4}x^{\frac{1}{2}}dx=2\times\frac{2}{3}x^{\frac{3}{2}}\big|_{0}^{4}=\frac{4}{3}\times(4)^{\frac{3}{2}}=\frac{32}{3}$$

Step2: Calculate \(\overline{x}\)

The formula for \(\overline{x}\) is \(\overline{x}=\frac{1}{A}\int_{a}^{b}x\cdot f(x)dx\).

$$
LATEXBLOCK0
$$

\(\overline{x}=\frac{3}{32}\times\frac{128}{5}=\frac{12}{5}\)

Step3: Calculate \(\overline{y}\)

The formula for \(\overline{y}\) is \(\overline{y}=\frac{1}{A}\int_{a}^{b}\frac{1}{2}[f(x)]^{2}dx\).

$$
LATEXBLOCK1
$$

\(\overline{y}=\frac{3}{32}\times16=\frac{3}{2}\)

Answer:

The centroid is \((\frac{12}{5},\frac{3}{2})\)