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.

3 questions · Moderate -0.9

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CAIE P1 2023 June Q5
4 marks Moderate -0.5
5 \includegraphics[max width=\textwidth, alt={}, center]{1662cb34-273c-461d-908c-9fe2ffe889b4-06_599_1086_274_518} The diagram shows the curve with equation \(y = 10 x ^ { \frac { 1 } { 2 } } - \frac { 5 } { 2 } x ^ { \frac { 3 } { 2 } }\) for \(x > 0\). The curve meets the \(x\)-axis at the points \(( 0,0 )\) and \(( 4,0 )\). Find the area of the shaded region.
The diagram shows a sector \(O A B\) of a circle with centre \(O\). Angle \(A O B = \theta\) radians and \(O P = A P = x\).
  1. Show that the arc length \(A B\) is \(2 x \theta \cos \theta\).
  2. Find the area of the shaded region \(A P B\) in terms of \(x\) and \(\theta\).
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 \(x = 4\).
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.