3. find the centroid of the planar region bounded by y = x², y = 0, and x = 1?

Explanation:

Step1: Recall centroid formulas

The centroid $(\overline{x},\overline{y})$ of a planar region bounded by $y = f(x)$, $y = 0$, $x=a$, and $x = b$ has formulas $\overline{x}=\frac{\int_{a}^{b}x\cdot f(x)dx}{\int_{a}^{b}f(x)dx}$ and $\overline{y}=\frac{\frac{1}{2}\int_{a}^{b}[f(x)]^{2}dx}{\int_{a}^{b}f(x)dx}$. Here $a = 0$, $b = 1$, and $f(x)=x^{2}$.

Step2: Calculate $\int_{a}^{b}f(x)dx$

We calculate $\int_{0}^{1}x^{2}dx$. Using the power - rule $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)$, we have $\int_{0}^{1}x^{2}dx=\left[\frac{x^{3}}{3}\right]_{0}^{1}=\frac{1}{3}$.

Step3: Calculate $\int_{a}^{b}x\cdot f(x)dx$

We calculate $\int_{0}^{1}x\cdot x^{2}dx=\int_{0}^{1}x^{3}dx$. By the power - rule, $\int_{0}^{1}x^{3}dx=\left[\frac{x^{4}}{4}\right]_{0}^{1}=\frac{1}{4}$.

Step4: Calculate $\overline{x}$

Using the formula $\overline{x}=\frac{\int_{a}^{b}x\cdot f(x)dx}{\int_{a}^{b}f(x)dx}$, we substitute the values: $\overline{x}=\frac{\frac{1}{4}}{\frac{1}{3}}=\frac{3}{4}$.

Step5: Calculate $\frac{1}{2}\int_{a}^{b}[f(x)]^{2}dx$

We calculate $\frac{1}{2}\int_{0}^{1}(x^{2})^{2}dx=\frac{1}{2}\int_{0}^{1}x^{4}dx$. By the power - rule, $\frac{1}{2}\int_{0}^{1}x^{4}dx=\frac{1}{2}\left[\frac{x^{5}}{5}\right]_{0}^{1}=\frac{1}{10}$.

Step6: Calculate $\overline{y}$

Using the formula $\overline{y}=\frac{\frac{1}{2}\int_{a}^{b}[f(x)]^{2}dx}{\int_{a}^{b}f(x)dx}$, we substitute the values: $\overline{y}=\frac{\frac{1}{10}}{\frac{1}{3}}=\frac{3}{10}$.

Answer:

The centroid is $\left(\frac{3}{4},\frac{3}{10}\right)$