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 $n > 1$.\\
(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 $S$.