4.08d Volumes of revolution: about x and y axes

387 questions

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OCR Further Pure Core 2 2018 September Q5
8 marks Challenging +1.2
The region \(R\) between the \(x\)-axis, the curve \(y = \frac{1}{\sqrt{p+x^3}}\) and the lines \(x = \sqrt{p}\) and \(x = \sqrt{3p}\), where \(p\) is a positive parameter, is rotated by \(2\pi\) radians about the \(x\)-axis to form a solid of revolution \(S\).
  1. Find and simplify an algebraic expression, in terms of \(p\), for the exact volume of \(S\). [5]
  2. Given that \(R\) must lie entirely between the lines \(x = 1\) and \(x = \sqrt{48}\) find in exact form
OCR Further Additional Pure 2018 September Q2
10 marks Challenging +1.8
In this question, you must show detailed reasoning. A curve is defined parametrically by \(x = t^3 - 3t + 1\), \(y = 3t^2 - 1\), for \(0 \leq t \leq 5\). Find, in exact form,
  1. the length of the curve, [6]
  2. the area of the surface generated when the curve is rotated completely about the \(x\)-axis. [4]
OCR Further Pure Core 2 2018 December Q8
7 marks Challenging +1.8
\includegraphics{figure_8} The figure shows part of the graph of \(y = (x - 3)\sqrt{\ln x}\). The portion of the graph below the \(x\)-axis is rotated by \(2\pi\) radians around the \(x\)-axis to form a solid of revolution, S. Determine the exact volume of S. [7]
OCR Further Additional Pure 2017 Specimen Q1
4 marks Challenging +1.2
A curve is given by \(x = t^2 - 2\ln t\), \(y = 4t\) for \(t > 0\). When the arc of the curve between the points where \(t = 1\) and \(t = 4\) is rotated through \(2\pi\) radians about the \(x\)-axis, a surface of revolution is formed with surface area \(A\). Given that \(A = k\pi\), where \(k\) is an integer, write down an integral which gives \(A\) and find the value of \(k\). [4]
Pre-U Pre-U 9794/2 2010 June Q9
15 marks Challenging +1.2
  1. Show that $$\int x^n \ln x \, dx = \frac{x^{n+1}}{(n+1)^2}\left((n+1)\ln a - 1\right) + \frac{1}{(n+1)^2},$$ where \(n \neq -1\) and \(a > 1\). [6]
    1. Determine the \(x\)-coordinate of the point of intersection of the curves \(y = x^3 \ln x\) and \(y = x \ln 2^x\), where \(x > 0\). [2]
    2. Find the exact value of the area of the region enclosed between these two curves, the line \(x = 1\) and their point of intersection. Express your answer in the form \(b + c \ln 2\), where \(b\) and \(c\) are rational. [4]
  3. The curve \(y = (x^3 \ln x)^{0.5}\), for \(1 < x < e\), is rotated through \(2\pi\) radians about the \(x\)-axis. Determine the value of the resulting volume of revolution, giving your answer correct to 4 significant figures. [3]
Pre-U Pre-U 9794/1 2011 June Q5
5 marks Standard +0.3
A circle has equation \(x^2 + y^2 = 16\). Find the volume generated when the region in the first quadrant which is bounded by the circle and the lines \(x = 1\) and \(x = 2\) is rotated through \(2\pi\) radians about the \(x\)-axis. [5]
Pre-U Pre-U 9795/1 2011 June Q11
15 marks Hard +2.3
  1. Let \(I_n = \int_0^{\frac{\pi}{2}} \sec^n t \, dt\) for positive integers \(n\). Prove that, for \(n \geqslant 2\), $$(n - 1)I_n = \frac{2^{n-2}}{(\sqrt{3})^{n-1}} + (n - 2)I_{n-2}.$$ [5]
  2. The curve with parametric equations \(x = \tan t\), \(y = \frac{1}{4}\sec^2 t\), for \(0 \leqslant t \leqslant \frac{1}{4}\pi\), is rotated through \(2\pi\) radians about the \(x\)-axis to form a surface of revolution of area \(S\). Show that \(S = \pi I_5\) and evaluate \(S\) exactly. [10]
Pre-U Pre-U 9794/2 2012 June Q10
12 marks Standard +0.3
    1. Find \(\int \frac{e^x}{1 + e^x} dx\). [2]
    2. Hence evaluate \(\int_0^{\ln 3} \frac{e^x}{1 + e^x} dx\), giving your answer in the form \(\ln k\), where \(k\) is an integer. [3]
    1. Using the substitution \(u = 1 + e^x\), find \(\int \left(\frac{e^x}{1 + e^x}\right)^2 dx\). [5]
    2. Hence find the exact volume of the solid of revolution generated when the curve given by \(y = \frac{e^x}{1 + e^x}\), between \(x = -\ln 3\) and \(x = \ln 3\), is rotated through \(2\pi\) radians about the \(x\)-axis. [2]
Pre-U Pre-U 9795/1 2013 November Q13
24 marks Hard +2.3
  1. Let \(I_n = \int_0^{\alpha} \cosh^n x \, dx\) for integers \(n \geqslant 0\), where \(\alpha = \ln 2\).
    1. Prove that, for \(n \geqslant 2\), \(nI_n = \frac{3 \times 5^{n-1}}{4^n} + (n-1)I_{n-2}\). [5]
    2. A curve has parametric equations \(x = 12 \sinh t + 4 \sinh^3 t\), \(y = 3 \cosh^4 t\), \(0 \leqslant t \leqslant \ln 2\). Find the length of the arc of this curve, giving your answer in the form \(a + b \ln 2\) for rational numbers \(a\) and \(b\). [8]
  2. The circle with equation \(x^2 + (y - R)^2 = r^2\), where \(r < R\), is rotated through one revolution about the \(x\)-axis to form a solid of revolution called a torus. By using suitable parametric equations for the circle, determine, in terms of \(\pi\), \(R\) and \(r\), the surface area of this torus. [11]
Pre-U Pre-U 9795/1 2015 June Q12
22 marks Challenging +1.8
Let \(I_n = \int_0^2 x^n \sqrt{1 + 2x^2} \, \text{d}x\) for \(n = 0, 1, 2, 3, \ldots\).
    1. Evaluate \(I_1\). [3]
    2. Prove that, for \(n \geqslant 2\), $$(2n + 4)I_n = 27 \times 2^{n-1} - (n - 1)I_{n-2}.$$ [6]
    3. Using a suitable substitution, or otherwise, show that $$I_0 = 3 + \frac{1}{\sqrt{2}} \ln(1 + \sqrt{2}).$$ [8]
  2. The curve \(y = \frac{1}{\sqrt{2}} x^2\), between \(x = 0\) and \(x = 2\), is rotated through \(2\pi\) radians about the \(x\)-axis to form a surface with area \(S\). Find the exact value of \(S\). [5]
Pre-U Pre-U 9795/1 2018 June Q12
15 marks Challenging +1.8
The curve \(C\) is given by \(y = \frac{1}{4}x^2 - \frac{1}{2}\ln x\) for \(2 \leq x \leq 8\).
  1. Find, in its simplest exact form, the length of \(C\). [5]
  2. When \(C\) is rotated through \(2\pi\) radians about the \(x\)-axis, a surface of revolution is formed. Show that the area of this surface is \(\pi(270 - 47\ln 2 - 2(\ln 2)^2)\). [10]
CAIE FP1 2013 November Q9
Challenging +1.8
9 The curve \(C\) has parametric equations $$x = t ^ { 2 } , \quad y = t - \frac { 1 } { 3 } t ^ { 3 } , \quad \text { for } 0 \leqslant t \leqslant 1 .$$ Find the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find the coordinates of the centroid of the region bounded by \(C\), the \(x\)-axis and the line \(x = 1\).