Standard +0.8 This is a multi-step integration problem requiring: (1) differentiation using chain rule to find the tangent equation, (2) finding where the tangent meets the x-axis, (3) calculating the area under the curve via integration with substitution, (4) calculating the triangular area, and (5) subtracting to find the shaded region. The integration of (3x-1)^4 requires substitution and careful handling of limits. While the individual techniques are standard C3 content, the multi-stage nature, need for exact answers, and geometric reasoning make this moderately challenging—above average but not exceptionally difficult.
\includegraphics{figure_7}
The diagram shows the curve with equation \(y = (3x - 1)^4\). The point P on the curve has coordinates \((1, 16)\) and the tangent to the curve at P meets the \(x\)-axis at the point Q. The shaded region is bounded by PQ, the \(x\)-axis and that part of the curve for which \(\frac{1}{3} \leqslant x \leqslant 1\). Find the exact area of this shaded region. [10]
\includegraphics{figure_7}
The diagram shows the curve with equation $y = (3x - 1)^4$. The point P on the curve has coordinates $(1, 16)$ and the tangent to the curve at P meets the $x$-axis at the point Q. The shaded region is bounded by PQ, the $x$-axis and that part of the curve for which $\frac{1}{3} \leqslant x \leqslant 1$. Find the exact area of this shaded region. [10]
\hfill \mbox{\textit{OCR C3 2010 Q7 [10]}}