Standard +0.3 This is a straightforward integration by substitution question with a given substitution. Students must apply the substitution u²=1-x, find dx in terms of du, change limits, and integrate a polynomial. While it requires careful algebraic manipulation, the method is standard C4 content with no novel problem-solving required, making it slightly easier than average.
5.
\includegraphics[max width=\textwidth, alt={}, center]{5840974b-b08a-4818-9a59-97b2d3ce9890-1_469_809_1777_484}
The diagram shows the curve with equation \(y = x \sqrt { 1 - x } , 0 \leq x \leq 1\).
Use the substitution \(u ^ { 2 } = 1 - x\) to show that the area of the region bounded by the curve and the \(x\)-axis is \(\frac { 4 } { 15 }\).
5.\\
\includegraphics[max width=\textwidth, alt={}, center]{5840974b-b08a-4818-9a59-97b2d3ce9890-1_469_809_1777_484}
The diagram shows the curve with equation $y = x \sqrt { 1 - x } , 0 \leq x \leq 1$.\\
Use the substitution $u ^ { 2 } = 1 - x$ to show that the area of the region bounded by the curve and the $x$-axis is $\frac { 4 } { 15 }$.\\
\hfill \mbox{\textit{OCR C4 Q5 [7]}}