Let $\ell$ be the line tangent to the graph of $y = x^{n}$ at the point $(1,1)$, where $n > 1$.
(a) Find $\displaystyle\int_{0}^{1} x^{n}\,dx$ in terms of $n$.
(b) Let $T$ be the triangular region bounded by $\ell$, the $x$-axis, and the line $x = 1$. Show that the area of $T$ is $\dfrac{1}{2n}$.
(c) Let $S$ be the region bounded by the graph of $y = x^{n}$, the line $\ell$, and the $x$-axis. Express the area of $S$ in terms of $n$ and determine the value of $n$ that maximizes the area of $S$.

Let $\ell$ be the line tangent to the graph of $y = x ^ { n }$ at the point $( 1,1 )$, where $n > 1$, as shown above.
(a) Find $\int _ { 0 } ^ { 1 } x ^ { n } d x$ in terms of $n$.
(b) Let $T$ be the triangular region bounded by $\ell$, the $x$-axis, and the line $x = 1$. Show that the area of $T$ is $\frac { 1 } { 2 n }$.
(c) Let $S$ be the region bounded by the graph of $y = x ^ { n }$, the line $\ell$, and the $x$-axis. Express the area of $S$ in terms of $n$ and determine the value of $n$ that maximizes the area of $S$.

(a) Draw a qualitatively accurate sketch of the unique bounded region R in the first quadrant that has maximum possible finite area with boundary described as follows. R is bounded below by the graph of $y = x^2 - x^3$, bounded above by the graph of an equation of the form $y = kx$ (where $k$ is some constant), and R is entirely enclosed by the two given graphs, i.e., the boundary of the region R must be a subset of the union of the given two graphs (so R does not have any points on its boundary that are not on these two graphs). Clearly mark the relevant point(s) on the boundary where the two given graphs meet and write the coordinates of every such point.
(b) Consider the solid obtained by rotating the above region R around $Y$-axis. Show how to find the volume of this solid by doing the following: Carefully set up the calculation with justification. Do enough work with the resulting expression to reach a stage where the final numerical answer can be found mechanically by using standard symbolic formulas of algebra and/or calculus and substituting known values in them. Do not carry out the mechanical work to get the final numerical answer.

A triangle has vertices $A$, $B$, $C$. A point $P$ is chosen on side $AB$, and lines through $P$ parallel to the other sides create smaller triangles $APQ$ and $BPR$ and a parallelogram $PQCR$. Find the minimum value of the maximum of the areas of triangles $APQ$ and $BPR$ as a fraction of the area of $ABC$.