1.08a Fundamental theorem of calculus: integration as reverse of differentiation

241 questions

Sort by: Default | Easiest first | Hardest first
Edexcel C1 Q10
10 marks Moderate -0.8
The curve \(C\) with equation \(y = f(x)\), \(x \neq 0\), passes through the point \((3, 7\frac{1}{2})\). Given that \(f'(x) = 2x + \frac{3}{x^2}\),
  1. find \(f(x)\). [5]
  2. Verify that \(f(-2) = 5\). [1]
  3. Find an equation for the tangent to \(C\) at the point \((-2, 5)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]
Edexcel C1 Q7
9 marks Moderate -0.3
The curve \(C\) has equation \(y = f(x)\), \(x \neq 0\), and the point \(P(2, 1)\) lies on \(C\). Given that $$f'(x) = 3x^2 - 6 - \frac{8}{x^3},$$
  1. find \(f(x)\). [5]
  2. Find an equation for the tangent to \(C\) at the point \(P\), giving your answer in the form \(y = mx + c\), where \(m\) and \(c\) are integers. [4]
Edexcel C1 Q32
4 marks Standard +0.3
A container made from thin metal is in the shape of a right circular cylinder with height \(h\) cm and base radius \(r\) cm. The container has no lid. When full of water, the container holds 500 cm³ of water. Show that the exterior surface area, \(A\) cm², of the container is given by $$A = \pi r^2 + \frac{1000}{r}.$$ [4]
Edexcel M3 2012 June Q1
9 marks Standard +0.3
A particle \(P\) is moving along the positive \(x\)-axis. At time \(t = 0\), \(P\) is at the origin \(O\). At time \(t\) seconds, \(P\) is \(x\) metres from \(O\) and has velocity \(v = 2e^{-t}\) m s\(^{-1}\) in the direction of \(x\) increasing.
  1. Find the acceleration of \(P\) in terms of \(x\). [3]
  2. Find \(x\) in terms of \(t\). [6]
Edexcel C1 Q9
11 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = \text{f}(x)\) which crosses the \(x\)-axis at the origin and at the points \(A\) and \(B\). Given that $$\text{f}'(x) = 6 - 4x - 3x^2,$$
  1. find an expression for \(y\) in terms of \(x\), [5]
  2. show that \(AB = k\sqrt{7}\), where \(k\) is an integer to be found. [6]
Edexcel C1 Q6
7 marks Moderate -0.5
Given that $$\frac{dy}{dx} = 3\sqrt{x} - x^2,$$ and that \(y = \frac{2}{3}\) when \(x = 1\), find the value of \(y\) when \(x = 4\). [7]
Edexcel C1 Q10
12 marks Moderate -0.3
The curve \(C\) with equation \(y = \text{f}(x)\) is such that $$\frac{\text{d}y}{\text{d}x} = 3x^2 + 4x + k,$$ where \(k\) is a constant. Given that \(C\) passes through the points \((0, -2)\) and \((2, 18)\),
  1. show that \(k = 2\) and find an equation for \(C\), [7]
  2. show that the line with equation \(y = x - 2\) is a tangent to \(C\) and find the coordinates of the point of contact. [5]
Edexcel C1 Q10
13 marks Moderate -0.3
The curve \(C\) has the equation \(y = f(x)\). Given that $$\frac{dy}{dx} = 8x - \frac{2}{x^3}, \quad x \neq 0,$$ and that the point \(P(1, 1)\) lies on \(C\),
  1. find an equation for the tangent to \(C\) at \(P\) in the form \(y = mx + c\), [3]
  2. find an equation for \(C\), [5]
  3. find the \(x\)-coordinates of the points where \(C\) meets the \(x\)-axis, giving your answers in the form \(k\sqrt{2}\). [5]
Edexcel C1 Q10
13 marks Moderate -0.3
\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = \text{f}(x)\). The curve meets the \(x\)-axis at the origin and at the point \(A\). Given that $$\text{f}'(x) = 3x^{\frac{1}{2}} - 4x^{-\frac{1}{2}},$$
  1. find f\((x)\). [5]
  2. Find the coordinates of \(A\). [2]
The point \(B\) on the curve has \(x\)-coordinate 2.
  1. Find an equation for the tangent to the curve at \(B\) in the form \(y = mx + c\). [6]
OCR C2 2007 January Q3
5 marks Easy -1.2
  1. Find \(\int (4x - 5) dx\). [2]
  2. The gradient of a curve is given by \(\frac{dy}{dx} = 4x - 5\). The curve passes through the point \((3, 7)\). Find the equation of the curve. [3]
OCR C2 Specimen Q2
6 marks Easy -1.2
  1. Find \(\int \frac{1}{x^2} dx\). [3]
  2. The gradient of a curve is given by \(\frac{dy}{dx} = \frac{1}{x^2}\). Find the equation of the curve, given that it passes through the point \((1, 3)\). [3]
OCR MEI C2 2006 June Q5
4 marks Moderate -0.8
The gradient of a curve is given by \(\frac{dy}{dx} = 3 - x^2\). The curve passes through the point \((6, 1)\). Find the equation of the curve. [4]
OCR MEI C2 2008 June Q6
4 marks Moderate -0.8
A curve has gradient given by \(\frac{\text{d}y}{\text{d}x} = 6\sqrt{x}\). Find the equation of the curve, given that it passes through the point \((9, 105)\). [4]
OCR MEI C2 2010 June Q6
5 marks Moderate -0.5
The gradient of a curve is \(6x^2 + 12x^{\frac{1}{2}}\). The curve passes through the point \((4, 10)\). Find the equation of the curve. [5]
OCR MEI C2 2013 June Q3
5 marks Moderate -0.8
The gradient of a curve is given by \(\frac{dy}{dx} = \frac{18}{x^3} + 2\). The curve passes through the point \((3, 6)\). Find the equation of the curve. [5]
OCR C2 Q9
13 marks Moderate -0.3
  1. Evaluate $$\int_1^3 (3 - \sqrt{x})^2 \, dx,$$ giving your answer in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are integers. [6]
  2. The gradient of a curve is given by $$\frac{dy}{dx} = 3x^2 + 4x + k,$$ where \(k\) is a constant. Given that the curve passes through the points \((0, -2)\) and \((2, 18)\), show that \(k = 2\) and find an equation for the curve. [7]
OCR C2 Q8
12 marks Moderate -0.3
  1. The gradient of a curve is given by $$\frac{dy}{dx} = 3 - \frac{2}{x^2}, \quad x \neq 0.$$ Find an equation for the curve given that it passes through the point \((2, 6)\). [6]
  2. Show that $$\int_2^3 (6\sqrt{x} - \frac{4}{\sqrt{x}}) \, dx = k\sqrt{3},$$ where \(k\) is an integer to be found. [6]
OCR MEI C2 Q2
5 marks Moderate -0.8
The gradient of a curve is given by \(\frac{dy}{dx} = \frac{6}{x^3}\). The curve passes through \((1, 4)\). Find the equation of the curve. [5]
OCR MEI C2 Q11
4 marks Moderate -0.8
A curve has gradient given by \(\frac{dy}{dx} = 6x^2 + 8x\). The curve passes through the point \((1, 5)\). Find the equation of the curve. [4]
OCR MEI C2 Q1
13 marks Moderate -0.3
The gradient of a curve is given by \(\frac{dy}{dx} = 4x + 3\). The curve passes through the point \((2, 9)\).
  1. Find the equation of the tangent to the curve at the point \((2, 9)\). [3]
  2. Find the equation of the curve and the coordinates of its points of intersection with the \(x\)-axis. Find also the coordinates of the minimum point of this curve. [7]
  3. Find the equation of the curve after it has been stretched parallel to the \(x\)-axis with scale factor \(\frac{1}{2}\). Write down the coordinates of the minimum point of the transformed curve. [3]
Edexcel C4 Q5
11 marks Standard +0.3
\includegraphics{figure_1} The curve \(C\) has equation \(y = f(x)\), \(x \in \mathbb{R}\). Figure 1 shows the part of \(C\) for which \(0 \leq x \leq 2\). Given that $$\frac{dy}{dx} = e^x - 2x^2,$$ and that \(C\) has a single maximum, at \(x = k\),
  1. show that \(1.48 < k < 1.49\). [3]
Given also that the point \((0, 5)\) lies on \(C\),
  1. find \(f(x)\). [4]
The finite region \(R\) is bounded by \(C\), the coordinate axes and the line \(x = 2\).
  1. Use integration to find the exact area of \(R\). [4]
OCR C4 2006 June Q3
8 marks Moderate -0.3
  1. Express \(\frac{3 - 2x}{x(3 - x)}\) in partial fractions. [3]
  2. Show that \(\int_1^2 \frac{3 - 2x}{x(3 - x)} dx = 0\). [4]
  3. What does the result of part (ii) indicate about the graph of \(y = \frac{3 - 2x}{x(3 - x)}\) between \(x = 1\) and \(x = 2\)? [1]
OCR M1 Specimen Q5
13 marks Moderate -0.3
A particle \(P\) moves in a straight line so that, at time \(t\) seconds after leaving a fixed point \(O\), its acceleration is \(-\frac{1}{10}t \text{ m s}^{-2}\). At time \(t = 0\), the velocity of \(P\) is \(V \text{ m s}^{-1}\).
  1. Find, by integration, an expression in terms of \(t\) and \(V\) for the velocity of \(P\). [4]
  2. Find the value of \(V\), given that \(P\) is instantaneously at rest when \(t = 10\). [2]
  3. Find the displacement of \(P\) from \(O\) when \(t = 10\). [4]
  4. Find the speed with which the particle returns to \(O\). [3]
OCR MEI M1 2008 January Q5
8 marks Standard +0.3
\includegraphics{figure_5} A toy car is moving along the straight line \(Ox\), where O is the origin. The time \(t\) is in seconds. At time \(t = 0\) the car is at A, 3 m from O as shown in Fig. 5. The velocity of the car, \(v\) m s\(^{-1}\), is given by $$v = 2 + 12t - 3t^2.$$ Calculate the distance of the car from O when its acceleration is zero. [8]
Edexcel M4 2004 January Q4
14 marks Challenging +1.8
A particle \(P\) of mass \(m\) is attached to the mid-point of a light elastic string, of natural length \(2L\) and modulus of elasticity \(2mk^2L\), where \(k\) is a positive constant. The ends of the string are attached to points \(A\) and \(B\) on a smooth horizontal surface, where \(AB = 3L\). The particle is released from rest at the point \(C\), where \(AC = 2L\) and \(ACB\) is a straight line. During the subsequent motion \(P\) experiences air resistance of magnitude \(2mkv\), where \(v\) is the speed of \(P\). At time \(t\), \(AP = 1.5L + x\).
  1. Show that \(\frac{d^2x}{dt^2} + 2k\frac{dx}{dt} + 4k^2x = 0\). [6]
  2. Find an expression, in terms of \(t\), \(k\) and \(L\), for the distance \(AP\) at time \(t\). [8]