Area involving fractional powers

A question is this type if and only if the curve equation involves fractional or negative powers of x (like x^(1/2), x^(-2)) requiring power rule integration.

10 questions · Moderate -0.4

1.08e Area between curve and x-axis: using definite integrals
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CAIE P1 2005 June Q9
10 marks Moderate -0.3
9 A curve has equation \(y = \frac { 4 } { \sqrt { } x }\).
  1. The normal to the curve at the point \(( 4,2 )\) meets the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\). Find the length of \(P Q\), correct to 3 significant figures.
  2. Find the area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 4\).
CAIE P1 2007 November Q2
4 marks Easy -1.8
2 Find the area of the region enclosed by the curve \(y = 2 \sqrt { } x\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\).
Edexcel C2 2016 June Q7
6 marks Moderate -0.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{582cda45-80fc-43a8-90e6-1cae08cb1534-12_563_812_244_630} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve with equation $$y = 3 x - x ^ { \frac { 3 } { 2 } } , \quad x \geqslant 0$$ The finite region \(S\), bounded by the \(x\)-axis and the curve, is shown shaded in Figure 3.
  1. Find $$\int \left( 3 x - x ^ { \frac { 3 } { 2 } } \right) \mathrm { d } x$$
  2. Hence find the area of \(S\).
OCR C2 Q2
5 marks Moderate -0.3
2. \includegraphics[max width=\textwidth, alt={}, center]{5025c118-e763-424b-b2c1-5452953a43a9-1_550_901_817_468} The diagram shows the curve with equation \(y = \sqrt { x } + \frac { 8 } { x ^ { 2 } } , x > 0\).
Show that the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 9\) is \(24 \frac { 4 } { 9 }\).
Edexcel C2 Q8
12 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{089f5506-94ac-489f-b219-e67fa6ca834f-4_536_883_248_486} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows the curve with equation \(y = \sqrt { x } + \frac { 8 } { x ^ { 2 } } , x > 0\).
  1. Find the coordinates of the minimum point of the curve.
  2. Show that the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 9\) is \(24 \frac { 4 } { 9 }\).
Edexcel C2 Q5
8 marks Moderate -0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{824da843-3ea1-4a83-9170-d0082b2e8d1c-3_458_862_906_511} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the curve with equation \(y = 4 x ^ { \frac { 1 } { 3 } } - x , x \geq 0\).
The curve meets the \(x\)-axis at the origin and at the point \(A\) with coordinates \(( a , 0 )\).
  1. Show that \(a = 8\).
  2. Find the area of the finite region bounded by the curve and the positive \(x\)-axis.
OCR MEI AS Paper 2 2024 June Q11
6 marks Moderate -0.8
  1. Verify that the curve cuts the \(x\)-axis at \(x = 4\) and at \(x = 9\). The curve does not cut or touch the \(x\)-axis at any other points.
  2. Determine the exact area bounded by the curve and the \(x\)-axis.
OCR PURE Q8
9 marks Standard +0.3
8 In this question you must show detailed reasoning. The diagram shows part of the graph of \(y = 2 x ^ { \frac { 1 } { 3 } } - \frac { 7 } { x ^ { \frac { 1 } { 3 } } }\). The shaded region is enclosed by the curve, the \(x\)-axis and the lines \(x = 8\) and \(x = a\), where \(a > 8\). \includegraphics[max width=\textwidth, alt={}, center]{efde7b10-b4f3-469f-ba91-b765a16ea835-5_577_1164_477_438} Given that the area of the shaded region is 45 square units, find the value of \(a\).
AQA AS Paper 2 2019 June Q6
5 marks Moderate -0.3
6 A curve has equation \(y = \frac { 2 } { x \sqrt { x } }\) \includegraphics[max width=\textwidth, alt={}, center]{b45dc98e-1699-47c9-9228-5abe0e5c9195-05_508_549_420_744} The region enclosed between the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = a\) has area 3 units. Given that \(a > 1\), find the value of \(a\).
Fully justify your answer.
OCR C2 Q6
8 marks Moderate -0.3
\includegraphics{figure_6} The diagram shows the curve with equation \(y = 4x^{\frac{1}{3}} - x\), \(x \geq 0\). The curve meets the \(x\)-axis at the origin and at the point \(A\) with coordinates \((a, 0)\).
  1. Show that \(a = 8\). [3]
  2. Find the area of the finite region bounded by the curve and the positive \(x\)-axis. [5]