Region bounded by two curves

A question is this type if and only if it asks to find the area between two curves (not curve and line), requiring integration of the difference between two functions.

5 questions · Moderate -0.1

1.08e Area between curve and x-axis: using definite integrals
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AQA C2 2009 January Q4
14 marks Moderate -0.3
4 The diagram shows a sketch of the curves with equations \(y = 2 x ^ { \frac { 3 } { 2 } }\) and \(y = 8 x ^ { \frac { 1 } { 2 } }\). \includegraphics[max width=\textwidth, alt={}, center]{0e19665b-5ee5-49e4-8de2-6c8dd17f61eb-3_433_720_1452_644} The curves intersect at the origin and at the point \(A\), where \(x = 4\).
    1. For the curve \(y = 2 x ^ { \frac { 3 } { 2 } }\), find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 4\).
      (2 marks)
    2. Find an equation of the normal to the curve \(y = 2 x ^ { \frac { 3 } { 2 } }\) at the point \(A\).
    1. Find \(\int 8 x ^ { \frac { 1 } { 2 } } \mathrm {~d} x\).
    2. Find the area of the shaded region bounded by the two curves.
  3. Describe a single geometrical transformation that maps the graph of \(y = 2 x ^ { \frac { 3 } { 2 } }\) onto the graph of \(y = 2 ( x + 3 ) ^ { \frac { 3 } { 2 } }\).
    (2 marks)
Pre-U Pre-U 9794/1 2012 Specimen Q9
10 marks Moderate -0.3
9
  1. On the same axes, sketch the curves \(y = 3 + 2 x - x ^ { 2 }\) and \(y = x + 1\).
  2. Find the exact area of the region contained between the curves \(y = 3 + 2 x - x ^ { 2 }\) and \(y = x + 1\).
CAIE P1 2024 November Q7
8 marks Standard +0.3
  1. By expressing \(-2x^2 + 8x + 11\) in the form \(-a(x - h)^2 + c\), where \(a\), \(b\) and \(c\) are positive integers, find the coordinates of the vertex of the graph with equation \(y = -2x^2 + 8x + 11\). [3]
  2. \includegraphics{figure_7} The diagram shows part of the curve with equation \(y = -2x^2 + 8x + 11\) and the line with equation \(y = 8x + 9\). Find the area of the shaded region. [5]
CAIE P1 2014 November Q9
10 marks Moderate -0.3
\includegraphics{figure_9} The diagram shows parts of the graphs of \(y = x + 2\) and \(y = 3\sqrt{x}\) intersecting at points \(A\) and \(B\).
  1. Write down an equation satisfied by the \(x\)-coordinates of \(A\) and \(B\). Solve this equation and hence find the coordinates of \(A\) and \(B\). [4]
  2. Find by integration the area of the shaded region. [6]
WJEC Unit 3 Specimen Q9
6 marks Standard +0.3
\includegraphics{figure_9} The diagram above shows a sketch of the curves \(y = x^2 + 4\) and \(y = 12 - x^2\). Find the area of the region bounded by the two curves. [6]