Further calculus questions

Questions to attempt

6 questions on Further calculus, each with its mark scheme and worked method. Reveal the answer when you have had a go.

1. CP-5.1 · Derive formulae for and calculate volumes of revolution.

   Q1

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   1.

   The region bounded by $x=y^2+1$, the $y$-axis, $y=0$ and $y=2$ is rotated through $2\pi$ radians about the $y$-axis. Find the exact volume.

   (4)

   (Total for Question 1 is 4 marks)

   Reveal answer and method

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2. CP-5.2 · Evaluate improper integrals where either the integrand is undefined at a value in the range of integration or the range of integration extends to infinity.

   Q2

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   2.

   Evaluate $\displaystyle \int_0^4\dfrac{1}{\sqrt{x}}\,dx$.

   (3)

   (Total for Question 2 is 3 marks)

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3. CP-5.3 · Understand and evaluate the mean value of a function.

   Q3

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   3.

   Determine the exact mean value of $f(x)=\dfrac{1}{x+1}$ on $0\leq x\leq e-1$.

   (4)

   (Total for Question 3 is 4 marks)

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4. CP-5.4 · Integrate using partial fractions.

   Q4

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   4.

   Find $\displaystyle \int\dfrac{2x+5}{(x+1)^2}\,dx$.

   (4)

   (Total for Question 4 is 4 marks)

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5. CP-5.5 · Differentiate inverse trigonometric functions.

   Q5

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   5.

   Differentiate $y=\arctan\left(\dfrac{x-1}{2}\right)$.

   (3)

   (Total for Question 5 is 3 marks)

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6. CP-5.6 · Integrate functions of the form (a^2 - x^2)^(-1/2) and (a^2 - x^2)^(-1) and be able to choose trigonometric substitutions to integrate associated functions.

   Q6

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   6.

   Find $\displaystyle \int\dfrac{1}{9-x^2}\,dx$.

   (3)

   (Total for Question 6 is 3 marks)

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CP-5 Further calculus revision notes