Integration: past paper questions

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• Question 8May/June 2026, paper 11Structured8 marksHas a diagram

  The diagram shows part of the curve with equation $y=\frac{5}{2x-1}$ and the line $y=7-2x$. Find the exact volume formed when the shaded region is rotated through ${360}^{\circ }$ about the $x$-axis.

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• Question 8Oct/Nov 2025, paper 11Structured7 marksHas a diagram

  The diagram shows the curve with equation $y=\frac{1}{2}\sqrt{x}$ and the point $P$ with coordinates $(9,\frac{3}{2})$. The shaded region is bounded by the curve and the lines $x=0$ and $y=\frac{3}{2}$. Find the area of the shaded region.

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• Question 5Oct/Nov 2024, paper 11Structured7 marks

  The equation of a curve is such that $\frac{\mathrm{d}y}{\mathrm{d}x}=4x-3\sqrt{x}+1$. Find the $x$-coordinate of the point on the curve at which the gradient is $\frac{11}{2}$.

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• Question 8Oct/Nov 2023, paper 11Structured9 marksHas a diagram

  The diagram shows the curves with equations $y=2(2x-3{)}^4$ and $y=(2x-3{)}^2+1$ meeting at points $A$ and $B$. By using the substitution $u=2x-3$ find, by calculation, the coordinates of $A$ and $B$.

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• Question 10May/June 2026, paper 11Structured11 marksHas a diagram

  The equation of a curve is such that $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}=3x^2-6$. The point $P(2,3)$ is a stationary point on the curve. By evaluating $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}$ at $P$, determine the nature of this stationary point.

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• Question 11Oct/Nov 2025, paper 11Structured11 marks

  A curve passes through the point $P(4,3)$ and is such that $$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{8}{x^2}-\frac{10}{(2x-3{)}^2}\mathrm{.}$$ Find the equation of the normal to the curve at $P$.

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• Question 7Oct/Nov 2024, paper 11Structured8 marks

  The diagram shows part of the curve with equation $y=\frac{12}{\sqrt[3]{2x+1}}$. The point $A$ on the curve has coordinates $(\frac{7}{2},6)$. Find the equation of the tangent to the curve at $A$.

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• Question 10Oct/Nov 2023, paper 11Structured11 marks

  A curve has a stationary point at $(2,-10)$ and is such that $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}=6x$. Find $\frac{\mathrm{d}y}{\mathrm{d}x}$.

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• Question 5May/June 2026, paper 12Structured7 marks

  The equation of a curve is such that $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}=6x^2+p$, where $p$ is a constant. The curve has minimum points at $(2,3)$ and $(-2,3)$. Find the coordinates of the other stationary point of the curve.

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• Question 4Oct/Nov 2025, paper 12Structured6 marks

  The equation of a curve is such that $\frac{\mathrm{d}y}{\mathrm{d}x}=k{x}^3+\frac{2}{x^2}$, where $k$ is a constant. The curve passes through the point $S(2,20)$ and the gradient of the curve at $S$ is $\frac{65}{2}$. Find the value of $k$.

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• Question 7Oct/Nov 2024, paper 12Structured8 marksHas a diagram

  By expressing $-2x^2+8x+11$ in the form $-a(x-b{)}^2+c$, where $a$, $b$ and $c$ are positive integers, find the coordinates of the vertex of the graph with equation $y=-2x^2+8x+11$.

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• Question 3Oct/Nov 2023, paper 12Structured6 marksHas a diagram

  The equation of a curve is such that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{2}x+\frac{72}{x^4}$. The curve passes through the point $P(2,8)$. Find the equation of the normal to the curve at $P$.

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• Question 9May/June 2026, paper 12Structured11 marksHas a diagram

  The diagram shows the curve with equation $y=3\sqrt{2x+1}$, the normal to the curve at the point $R(4,9)$ and the line $y=a$, where $a$ is a positive constant. Find the area of the shaded region.

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• Question 5Oct/Nov 2025, paper 12Structured8 marksHas a diagram

  The equation of a curve is $y=4x^{\frac{1}{2}}-x$. The curve has a maximum point when $x=a$ and crosses the $x$-axis at the point with coordinates $(b,0)$, where $b>0$. The shaded region is bounded by the curve, the line $x=a$ and the $x$-axis (see diagram).

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• Question 10Oct/Nov 2024, paper 12Structured10 marks

  A function $\mathrm{f}$ with domain $x>0$ is such that ${\mathrm{f}}^{\mathrm{\prime }}(x)=8(2x-3{)}^{\frac{1}{3}}-10x^{\frac{2}{3}}$. It is given that the curve with equation $y=\mathrm{f}(x)$ passes through the point $(1,0)$.

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• Question 9Oct/Nov 2023, paper 12Structured9 marksHas a diagram

  The diagram shows curves with equations $y=2x^{\frac{1}{2}}+13x^{-\frac{1}{2}}$ and $y=3x^{-\frac{1}{2}}+12$. The curves intersect at points $A$ and $B$. Find the coordinates of $A$ and $B$.

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• Question 8May/June 2026, paper 13Structured5 marksHas a diagram

  The diagram shows part of the curve with equation $y=\frac{2}{5x^3}$. The shaded region is bounded by the curve, the $x$-axis and the lines $x=1$ and $x=t$. It is given that the area of the shaded region is $\frac{1}{9}$.

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• Question 9Oct/Nov 2025, paper 13Structured8 marks

  The diagram shows part of the curve with equation $y=\frac{1}{2}x+\frac{4}{x}$ and the line $y=4.5$. Find the exact volume of the solid formed when the shaded region is rotated through ${360}^{\circ }$ about the $x$-axis.

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• Question 9Oct/Nov 2024, paper 13Structured7 marks

  The diagram shows the curves with equations $y=x^3-3x+3$ and $y=2x^3-4x^2+3$. Find the $x$-coordinates of the points of intersection of the curves.

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• Question 1Oct/Nov 2023, paper 13Structured4 marks

  A curve is such that its gradient at a point $(x,y)$ is given by $\frac{\mathrm{d}y}{\mathrm{d}x}=x-3x^{-\frac{1}{2}}$. It is given that the curve passes through the point $(4,1)$. Find the equation of the curve.

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• Question 2May/June 2025, paper 11Structured6 marks

  The equation of a curve is such that $\frac{\mathrm{d}y}{\mathrm{d}x}=4(2x-5{)}^3-9x^{\frac{1}{2}}$. The curve passes through the point $A(4,-\frac{11}{2})$. Find the gradient of the normal to the curve at the point $A$.

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• Question 9May/June 2024, paper 11Structured6 marksHas a diagram

  The diagram shows part of the curve with equation $y=\frac{1}{(5x-4{)}^{\frac{1}{3}}}$ and the lines $x=2.4$ and $y=1$. The curve intersects the line $y=1$ at the point $(1,1)$.

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• Question 11Oct/Nov 2023, paper 13Structured10 marksHas a diagram

  The diagram shows part of the curve with equation $y=x+\frac{2}{(2x-1{)}^2}$. The lines $x=1$ and $x=2$ intersect the curve at $P$ and $Q$ respectively and $R$ is the stationary point on the curve.

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• Question 4May/June 2025, paper 11Structured5 marksHas a diagram

  The diagram shows the curve with equation $y=5x^{\frac{3}{2}}-20x$ and the line with equation $y=x-16$. The $x$-coordinates of the points of intersection of the curve and line are 1 and 16. Find the area of the shaded region between the curve and the line.

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• Question 6May/June 2024, paper 12Structured9 marksHas a diagram

  The curve with equation $y=2x-8x^{\frac{1}{2}}$ has a minimum point at $A$ and intersects the positive $x$-axis at $B$. Find the coordinates of $A$ and $B$.

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• Question 10May/June 2023, paper 11Structured11 marksHas a diagram

  The diagram shows part of the curve with equation $y=\frac{4}{(2x-1{)}^2}$ and parts of the lines $x=1$ and $y=1$. The curve passes through the points $A(1,4)$ and $B(\frac{3}{2},1)$.

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• Question 11May/June 2023, paper 11Structured9 marks

  The equation of a curve is such that $\frac{\mathrm{d}y}{\mathrm{d}x}=6x^2-30x+6a$, where $a$ is a positive constant. The curve has a stationary point at $(a,-15)$. Find the value of $a$.

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• Question 9May/June 2025, paper 12Structured12 marks

  The equation of a curve is such that $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}=-\frac{24}{x^3}$. It is given that the curve has a stationary point at $(-2,19)$. Find an expression for $\frac{\mathrm{d}y}{\mathrm{d}x}$.

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• Question 6May/June 2024, paper 13Structured7 marks

  A curve passes through the point $(\frac{4}{5},-3)$ and is such that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{-20}{(5x-3{)}^2}$. Find the equation of the curve.

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• Question 1May/June 2023, paper 12Structured3 marks

  The equation of a curve is such that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{4}{(x-3{)}^3}$ for $x>3$. The curve passes through the point $(4,5)$. Find the equation of the curve.

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