Given the region bounded by $y=x^2$, $x=0$, and $y=4$, determine by direct integration the x-coordinate of the centroid of the area.

What is the average (mean) value of the function $3t^3-t^2$ over the interval $1\le t\le 2$?

For the function shown below, the definite integral from $1$ to $0$ of $f\left(x\right)$ $dx$, that is, ${\int }_0^{1}f\left(x\right)dx$, is closest to which of the following values?

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 $g\left(x\right)$?

Suppose the graph of $f$ consists of two regions between $x=0$ and $x=4$: from $x=0$ to $x=2$, $f\left(x\right)$ forms a triangle above the $x$-axis with area $3$; from $x=2$ to $x=4$, $f\left(x\right)$ forms a rectangle below the $x$-axis with area $4$. What is the value of the definite integral ${\int }_0^{4}f\left(x\right)dx$?

Find the length of the curve given by $r\left(t\right)=(8t,t^2,\frac{t^3}{12})$ for $0\le t\le 1$.

Evaluate the surface integral ${\int }_S^{}(x^{2}z+y^{2}z) dS$, where $S$ is the hemisphere $x^2+y^2+z^2=4$, $z\ge 0$.

2. Evaluate the definite integral from $1$ to $e$: ${\int }_1^{e}xln\left(x\right)dx$.

Evaluate the definite integral: ${\int }_0^3(1+7x) dx$.