Area under curve

Find the exact area of a region bounded by a curve and coordinate axes or lines using definite integration.

3 questions · Moderate -0.6

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Edexcel C2 2015 June Q6
9 marks Moderate -0.3
6. (a) Find $$\int 10 x \left( x ^ { \frac { 1 } { 2 } } - 2 \right) \mathrm { d } x$$ giving each term in its simplest form. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8a7593c3-4f0b-4351-afae-7bd98cfc351d-10_401_1002_543_470} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = 10 x \left( x ^ { \frac { 1 } { 2 } } - 2 \right) , \quad x \geqslant 0$$ The curve \(C\) starts at the origin and crosses the \(x\)-axis at the point \(( 4,0 )\). The area, shown shaded in Figure 2, consists of two finite regions and is bounded by the curve \(C\), the \(x\)-axis and the line \(x = 9\) (b) Use your answer from part (a) to find the total area of the shaded regions.
Edexcel C4 Q6
13 marks
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{a5902f63-b19f-4e37-94b8-35c3b47ab9de-10_579_1326_268_423}
\end{figure} Figure 2 shows the curve with equation $$y = x ^ { 2 } \sin \left( \frac { 1 } { 2 } x \right) , \quad 0 < x \leq 2 \pi .$$ The finite region \(R\) bounded by the line \(x = \pi\), the \(x\)-axis, and the curve is shown shaded in Fig 2.
  1. Find the exact value of the area of \(R\), by integration. Give your answer in terms of \(\pi\). The table shows corresponding values of \(x\) and \(y\).
    \(x\)\(\pi\)\(\frac { 5 \pi } { 4 }\)\(\frac { 3 \pi } { 2 }\)\(\frac { 7 \pi } { 4 }\)\(2 \pi\)
    \(y\)9.869614.24715.702\(G\)0
  2. Find the value of \(G\).
  3. Use the trapezium rule with values of \(x ^ { 2 } \sin \left( \frac { 1 } { 2 } x \right)\)
    1. at \(x = \pi , x = \frac { 3 \pi } { 2 }\) and \(x = 2 \pi\) to find an approximate value for the area \(R\), giving your answer to 4 significant figures,
    2. at \(x = \pi , x = \frac { 5 \pi } { 4 } , x = \frac { 3 \pi } { 2 } , x = \frac { 7 \pi } { 4 }\) and \(x = 2 \pi\) to find an improved approximation for the area \(R\), giving your answer to 4 significant figures.
      6. continued
AQA C1 2007 January Q6
14 marks Moderate -0.8
6 The curve with equation \(y = 3 x ^ { 5 } + 2 x + 5\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{33da89e2-f74f-4d5a-8bbd-ceaa728b6c34-5_428_563_372_740} The curve cuts the \(x\)-axis at the point \(A ( - 1,0 )\) and cuts the \(y\)-axis at the point \(B\).
    1. State the coordinates of the point \(B\) and hence find the area of the triangle \(A O B\), where \(O\) is the origin.
    2. Find \(\int \left( 3 x ^ { 5 } + 2 x + 5 \right) \mathrm { d } x\).
    3. Hence find the area of the shaded region bounded by the curve and the line \(A B\).
    1. Find the gradient of the curve with equation \(y = 3 x ^ { 5 } + 2 x + 5\) at the point \(A ( - 1,0 )\).
    2. Hence find an equation of the tangent to the curve at the point \(A\).