Cambridge OCR A Level · Further Mathematics B (MEI) - H645

Calculus: Practice Questions

5 multiple-choice questions marked as you go, and 5 written questions with worked solutions. All on Calculus.

10 questions28 marksFree, no account

Question 1

1 mark

Find the mean value of the function \(f(x) = x^2\) over the interval \([1, 4]\).

Question 2

1 mark

Determine the exact value of the mean value of the function \(f(x) = \frac{1}{\sqrt{9 - x^2}}\) over the interval \([0, 1.5]\).

Question 3

1 mark

Determine the exact value of the improper integral:
\(\int_{0}^{\infty} \frac{1}{(x+1)(x^2+1)} \, dx\)

Question 4

1 mark

Find the volume of the solid generated when the region bounded by the curve \(y = e^x\), the \(x\)-axis, and the lines \(x = 0\) and \(x = 1\) is rotated through \(2\pi\) radians about the \(x\)-axis.

Question 5

1 mark

A solid is generated by rotating the region bounded by the curve \(y = \cosh x\), the \(x\)-axis, and the lines \(x = 0\) and \(x = \ln 2\) through \(2\pi\) radians about the \(x\)-axis.
Which of the following expressions represents the volume of this solid?

Question 6

2 marks

Determine the mean value of the function \( f(x) = 3x^2 \) on the interval \( [1, 3] \).

Write your answer out first, then check it against the worked solution.

Question 7

3 marks

Calculate the exact mean value of the function \(f(x) = \text{sech}^2 x\) over the interval \([0, \ln 3]\).

Write your answer out first, then check it against the worked solution.

Question 8

6 marks

Find the exact volume of the solid generated when the region bounded by the curve \( y = \sqrt{\arctan x} \), the \( x \)-axis, and the line \( x = 1 \) is rotated through \( 2\pi \) radians about the \( x \)-axis.

Write your answer out first, then check it against the worked solution.

Question 9

4 marks

Consider the function \( f(x) = \frac{1}{x^2+4} \) for \( 0 \le x \le 2 \).
(a) Find the mean value of \( f(x) \) over the interval \( [0, 2] \). Give your answer in terms of \( \pi \). [3]
(b) The region bounded by the curve \( y = f(x) \), the \( x \)-axis, and the lines \( x = 0 \) and \( x = 2 \) is rotated through \( 2\pi \) radians about the \( x \)-axis. Write down, but do not evaluate, an integral expression for the volume of the solid generated. [1]

Write your answer out first, then check it against the worked solution.

Question 10

8 marks

(a) Use the substitution \( x = 2\sin \theta \) to show that \( \int \frac{1}{(4-x^2)^{\frac{3}{2}}} dx = \frac{x}{4\sqrt{4-x^2}} + C \). [3]
(b) Hence, evaluate the improper integral \( \int_{0}^{k} \frac{1}{(4-x^2)^{\frac{3}{2}}} dx \) in terms of \( k \), where \( 0 < k < 2 \), and show that the integral does not converge as \( k \to 2 \). [2]
(c) Find the exact mean value of the function \( g(x) = \frac{1}{x^2+2x+5} \) over the interval \( [-1, 1] \). [3]

Write your answer out first, then check it against the worked solution.

* The content provided by thinka is generated by AI and may not always be accurate or up-to-date. Please use it as a supplementary resource and verify with official materials.

You've seen the model answer. Now get yours marked.

This page can show you how a good answer looks. It cannot tell you what your answer was missing. thinka marks your written work against the real mark scheme in about 15 seconds.

Get My Answers Marked Back to the notes

Want more questions like these? Get a fresh set on this topic, marked as you go.