Shaded region area with quadratic

A question is this type if and only if it involves finding the area between a quadratic curve and a line using integration.

2 questions · Standard +0.3

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OCR MEI C2 Q1
12 marks Standard +0.3
1 Fig. 12 is a sketch of the curve \(y = 2 x ^ { 2 } - 11 x + 12\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{44b860fb-040f-4d3f-94d8-42eae41c0e2d-1_468_940_285_830} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure}
  1. Show that the curve intersects the \(x\)-axis at \(( 4,0 )\) and find the coordinates of the other point of intersection of the curve and the \(x\)-axis.
  2. Find the equation of the normal to the curve at the point \(( 4,0 )\). Show also that the area of the triangle bounded by this normal and the axes is 1.6 units \({ } ^ { 2 }\).
  3. Find the area of the region bounded by the curve and the \(x\)-axis.
AQA C1 2012 June Q5
13 marks Standard +0.3
5
    1. Express \(x ^ { 2 } - 3 x + 5\) in the form \(( x - p ) ^ { 2 } + q\).
    2. Hence write down the equation of the line of symmetry of the curve with equation \(y = x ^ { 2 } - 3 x + 5\).
  2. The curve \(C\) with equation \(y = x ^ { 2 } - 3 x + 5\) and the straight line \(y = x + 5\) intersect at the point \(A ( 0,5 )\) and at the point \(B\), as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{dbc25177-4a28-480f-93d5-41acb2a2d28c-4_471_707_653_676}
    1. Find the coordinates of the point \(B\).
    2. Find \(\int \left( x ^ { 2 } - 3 x + 5 \right) \mathrm { d } x\).
    3. Find the area of the shaded region \(R\) bounded by the curve \(C\) and the line segment \(A B\).