Find the area of the region bounded by the graph of $f\left(x\right)=\sin x$ and the $x$-axis on the interval $[0,\pi ]$.

Let $L\left(d\right)$ be the arc length of the curve $g\left(x\right)=3x^2$ from $x=0$ to $x=d$, where $d\ge 0$. Find an explicit expression for $L\left(d\right)$.

Determine the surface area generated by revolving the curve $y=4x$, for $0\le x\le 1$, about the $y$-axis.

Let $T$ be the region bounded above by $y=\sin \left(x\right)$ and below by $y=0$ for $0\le x\le \pi$. When revolving $T$ about the $y$-axis, why is using the washer method to compute the volume not straightforward?

Determine the volume of the solid whose base is the region $R$ bounded by the semicircle $y=\sqrt{4-x^2}$ and the $x$-axis, and whose cross sections perpendicular to the $x$-axis are equilateral triangles. Use the general slicing method.