Consider the function $f ( x ) = x \sin ^ { 2 } x$ on the interval $0 \leqq x \leqq \pi$. Let $\ell$ be the tangent to the curve $y = f ( x )$ that passes through the origin, where $\ell$ is not the $x$-axis. We are to find the area $S$ of the region bounded by the curve $y = f ( x )$ and the tangent $\ell$.
(1) For each of $\mathbf{A}$ $\sim$ $\mathbf{D}$ in the following sentences, choose the correct answer from among (0) $\sim$ (9) below.
When we denote the point of tangency of the curve $y = f ( x )$ and the tangent $\ell$ by $( t , f ( t ) )$, we have the equality $\mathbf{A}$, since $\ell$ passes through the origin. Further, since
$$f ^ { \prime } ( t ) = \mathbf { B } + 2 t \, \mathbf { C }$$
the $x$-coordinate of the point of tangency is $t = \mathbf { D }$.
(0) $f ( t ) = t f ^ { \prime } ( t )$ (1) $f ^ { \prime } ( t ) = t f ( t )$ (2) $\sin t$ (3) $\sin ^ { 2 } t$ (4) $\cos ^ { 2 } t$ (5) $\sin t \cos t$ (6) $\frac { \pi } { 2 }$ (7) $\frac { \pi } { 3 }$ (8) $\frac { \pi } { 4 }$ (9) $\frac { \pi } { 6 }$
(2) For each of $\mathbf { E }$ $\sim$ $\mathbf { G }$ in the following sentences, choose the correct answer from among (0) $\sim$ (9) below.
The antiderivative of the function $f ( x )$ is
$$\int f ( x ) d x = \mathbf { E } \left( 2 x ^ { 2 } - 2 x \mathbf { F } - \mathbf { G } \right) + C ,$$
where $C$ is the integral constant.
(0) $\frac { 1 } { 8 }$ (1) $\frac { 1 } { 4 }$ (2) $\frac { 1 } { 2 }$ (3) 2 (4) 4 (5) 8 (6) $\sin x$ (7) $\cos x$ (8) $\sin 2 x$ (9) $\cos 2 x$
(3) Thus, the area $S$ of the region bounded by the curve $y = f ( x )$ and the tangent $\ell$ is
$$S = \frac { \mathbf { H } } { \mathbf { I J } } \pi ^ { \mathbf { K } } - \frac { \mathbf { L } } { \mathbf{M} } .$$