Given the function $f\left(x\right)=x^2$ defined on the interval $0\le x\le 2$, determine the y-coordinate of the centroid of the region bounded by the graph of $f\left(x\right)$, the x-axis, and the lines $x=0$ and $x=2$.

Consider the geometric series with first term $2$ and common ratio $\frac{1}{2}$. Write out the first four terms of the series: $2,1,\frac{1}{2},\frac{1}{4}$. What is the sum of the infinite series?

Is the constant function $y\left(t\right)=-4$ a solution to the differential equation $y^{\prime }=t^2(y+4)$?

If $f\left(x\right)=x^2$, which of the following is the value of $f(-3)$?

If $f\left(x\right)=x^3$, which of the following is the value of $f(-2)$?

Let $g\left(x\right)={\int }_0^{x}f\left(t\right)dt$, where $f$ is the function whose graph is shown. Which of the following statements is true about $g\left(x\right)$?

Find the exact length of the curve given by $x=\frac{y^4}{8}+\frac{1}{4y^2}$, for $1\le y\le 3$.

Find the length of the curve given by $r\left(t\right)=(6t,t^2,\frac{t^3}{9})$ for $0\le t\le 1$.

Evaluate the iterated integral by converting to polar coordinates: $${\int }_0^2{\int }_{2-x^2}^{4-x^2}{x}^2+y^2 dy dx$$