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 $S$.
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 $S$.