Further calculus – further maths test to download as PDF

(a) Evaluate \(\displaystyle\int_0^\infty6e^{-2x}dx\). [2]

(b) Show that \(\displaystyle\int_1^\infty\frac1{\sqrt x}\,dx\) diverges. [2]

(a) Show that \(\displaystyle\int_0^1\frac{dx}{x^2+2x+2}=\arctan2-\dfrac\pi4\). [3]

(b) Evaluate \(\displaystyle\int_0^1\frac{dx}{\sqrt{4-x^2}}\) exactly. [2]

The region \(R\) is bounded by \(y=\dfrac1{\sqrt{1+x^2}}\), the \(x\)-axis, the \(y\)-axis and the line \(x=1\).

1. Show that the volume formed when \(R\) is rotated about the \(x\)-axis is \(\dfrac{\pi^2}4\). [3]
2. Find the area of the cross-section of the solid at \(x=1\) and the volume of a cylinder of that radius and length 1. [2]

Find the mean value of \(f(x)=4x-x^2\) over the interval \([0,4]\). [3]

A planter is made by rotating the curve \(y=\dfrac{x^2}4\) about the \(y\)-axis, where \(x\) and \(y\) are in centimetres, up to a depth of 9 cm.

1. Find the radius of the top of the planter. [1]
2. Find the volume of compost it holds, to the nearest cm\(^3\). [2]