Exact area with surds

A question is this type if and only if it requires finding an exact area that must be expressed in surd form (containing square roots) rather than decimals.

5 questions · Moderate -0.1

1.08e Area between curve and x-axis: using definite integrals
Sort by: Default | Easiest first | Hardest first
Edexcel AS Paper 1 2023 June Q5
5 marks Moderate -0.8
  1. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ce4f8375-0d88-4e48-85de-35f7e90b014d-10_488_519_365_772} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The finite region \(R\), shown shaded in Figure 2, is bounded by the curve with equation \(y = 4 x ^ { 2 } + 3\), the \(y\)-axis and the line with equation \(y = 23\) Show that the exact area of \(R\) is \(k \sqrt { 5 }\) where \(k\) is a rational constant to be found.
Edexcel PMT Mocks Q8
5 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d37eaba2-0a25-4abf-b2c8-1e08673229fb-14_1090_1205_274_456} \captionsetup{labelformat=empty} \caption{Figure 2
Figure 2 shows a sketch of part of the curve with equation $$y = \frac { 12 x - x ^ { 2 } } { \sqrt { x } } , \quad x > 0$$ The region \(R\), shows shaded in figure 2, is bounded by the curve, the line with equation \(x = 4\), the \(x\)-axis and the line with equation \(x = 8\).
Show that the area of the shaded region \(R\) is \(\frac { 128 } { 5 } ( 3 \sqrt { 2 } - 2 )\).}
\end{figure} (5)
Edexcel Paper 2 2022 June Q8
6 marks Standard +0.3
  1. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{824d73c5-525c-4876-ad66-33c8f1664277-18_633_730_386_669} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of a curve with equation $$y = \frac { ( x - 2 ) ( x - 4 ) } { 4 \sqrt { x } } \quad x > 0$$ The region \(R\), shown shaded in Figure 3, is bounded by the curve and the \(x\)-axis.
Find the exact area of \(R\), writing your answer in the form \(a \sqrt { 2 } + b\), where \(a\) and \(b\) are constants to be found.
OCR MEI AS Paper 2 Specimen Q3
5 marks Moderate -0.8
3 Show that the area of the region bounded by the curve \(y = 3 x ^ { - \frac { 3 } { 2 } }\), the lines \(x = 1 , x = 3\) and the \(x\)-axis is \(6 - 2 \sqrt { 3 }\).
OCR C3 2013 January Q5
9 marks Standard +0.3
\includegraphics{figure_5} The diagram shows the curve \(y = \frac{6}{\sqrt{3x + 1}}\). The shaded region is bounded by the curve and the lines \(x = 2\), \(x = 9\) and \(y = 0\).
  1. Show that the area of the shaded region is \(4\sqrt{7}\) square units. [4]
  2. The shaded region is rotated completely about the \(x\)-axis. Show that the volume of the solid produced can be written in the form \(k\ln 2\), where the exact value of the constant \(k\) is to be determined. [5]