Integration and the fundamental theorem of calculus – university Year 1 maths test to download as PDF

1. Find \(\displaystyle\int x\cos x\,dx\). (2 marks)
2. Evaluate \(\displaystyle\int_0^{1}\dfrac{x}{1+x^2}dx\). (2 marks)

Evaluate \(\displaystyle\int_2^3\dfrac{dx}{(x-1)(x+1)}\), giving the answer as a single logarithm.

The region \(R\) is bounded by \(y=x^2\), the \(x\)-axis and the line \(x=2\).

1. Find the area of \(R\). (1 mark)
2. Find the volume when \(R\) is rotated through \(2\pi\) about the \(x\)-axis. (2 marks)
3. Find the mean value of \(y=x^2\) on \([0,2]\). (1 mark)

1. Determine whether \(\displaystyle\int_1^\infty\dfrac{dx}{x^{3/2}}\) converges and find its value. (2 marks)
2. Show that \(\displaystyle\int_1^\infty\dfrac{dx}{\sqrt x}\) diverges. (2 marks)

1. Find \(\dfrac d{dx}\displaystyle\int_1^{x^3}\ln t\,dt\). (2 marks)
2. Estimate \(\displaystyle\int_0^1x^2dx\) using the right-endpoint Riemann sum with 4 strips. (2 marks)