Area between two curves

Find the exact area of a region bounded by two curves (neither being a non-horizontal straight line), requiring integration of the difference of two functions.

3 questions · Moderate -0.3

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CAIE P1 2023 November Q9
9 marks Standard +0.3
\includegraphics{figure_9} The diagram shows curves with equations \(y = 2x^{\frac{1}{2}} + 13x^{-\frac{1}{2}}\) and \(y = 3x^{-\frac{1}{4}} + 12\). The curves intersect at points \(A\) and \(B\).
  1. Find the coordinates of \(A\) and \(B\). [4]
  2. Hence find the area of the shaded region. [5]
CAIE P1 2024 November Q9
7 marks Moderate -0.3
\includegraphics{figure_9} The diagram shows the curves with equations \(y = x^3 - 3x + 3\) and \(y = 2x^3 - 4x^2 + 3\).
  1. Find the \(x\)-coordinates of the points of intersection of the curves. [3]
  2. Find the area of the shaded region. [4]
OCR C2 Specimen Q7
9 marks Moderate -0.8
\includegraphics{figure_7} The diagram shows the curves \(y = -3x^2 - 9x + 30\) and \(y = x^2 + 3x - 10\).
  1. Verify that the curves intersect at the points \(A(-5, 0)\) and \(B(2, 0)\). [2]
  2. Show that the area of the shaded region between the curves is given by \(\int_{-5}^{2} (-4x^2 - 12x + 40) dx\). [2]
  3. Hence or otherwise show that the area of the shaded region between the curves is \(228\frac{2}{3}\). [5]