Decreasing functions, improper integrals, normals, and volumes of revolution
Watch the whole lecture free
Make a free account to watch every lecture to the end: 5 practice questions, 2 real past-paper questions with the answers explained, the mistakes that lose marks, the full-marks tip, a one-card summary and a quick quiz.
That was the teaching part. The rest of this lecture is free with an account: 5 practice questions, 2 real past-paper questions with the answers explained, the mistakes that lose marks, the full-marks tip, a one-card summary and a quick quiz.
No password needed now: we email you a link to set one. Already have an account? Sign in, then come back to this page.
The teaching part (2 min) is free to watch. A free account opens the whole lecture: 5 practice questions, 2 real past-paper questions with the answers explained, the mistakes that lose marks, the full-marks tip, a one-card summary and a quick quiz.
What this lecture covers
- Integration at full stretch
- Where is it decreasing?
- Then find f
- A past-paper example
Read the transcript of the teaching part
Last time
Before we start: last time, in Definite integrals and plane areas, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.
Integration at full stretch
Everything you know about integration, in the five types that fill the end of Paper 1. Find where a function falls, then integrate back with a given point. Integrate all the way to infinity. Find a normal, then find the curve. And spin a region around an axis to make a solid, the volume of revolution, either about the x axis or the y axis.
Where is it decreasing?
A function decreases where its derivative is negative. Given f prime of x as 6 times 2x minus 3 squared, minus 6x, set it less than zero. Expand: 24 x squared minus 78 x plus 54. Divide by 6: 4 x squared minus 13 x plus 9. It factorises as x minus 1 times 4x minus 9. A positive quadratic is negative between its roots. So f is decreasing for 1 less than x less than 9 over 4.
Then find f
Integrate f prime: 6 times 2x minus 3 squared gives 2x minus 3 cubed. And 6x gives 3 x squared. Given f of 1 is minus 1: minus 1 minus 3 plus c equals minus 1. So c is 3. So f of x is 2x minus 3 cubed, minus 3 x squared, plus 3.
A past-paper example
May June 2024, Paper 12, question 9. A function f has derivative 6 times 2x minus 3 squared, minus 6x. Determine the set of values of x for which f is decreasing, and, given that f of 1 is minus 1, find f of x. The answers: 1 less than x less than 9 over 4, and f of x equals 2x minus 3 cubed, minus 3 x squared, plus 3.
How often it comes up
How often Cambridge has asked each part, counted by ExamYard from every past paper in its question bank: Volume of revolution about the x-axis.
The whole chapter: Integration.
Chapter 1.8: Integration
- Integration: powers of x and the reverse of differentiation 5 min
- Definite integrals and plane areas 5 min
- Decreasing functions, improper integrals, normals, and volumes of revolution 7 min
Also for A Level Maths 9709 Paper 1
Which topics come up most in A Level Maths 9709 Paper 1 past papers, counted from every paper since 2004.