Censai
PricingLog inSign up

Maths tutoring that adapts to you.

Personalised practice that builds your own custom Learning Profile - made for HSC students.

xyz1. Graph y = f(x)
Sign Up Free

Trusted by Students from Top High Schools

Abbotsleigh
Baulkham Hills High School
Hornsby Girls High School
James Ruse Agricultural High School
Normanhurst Boys High School
North Sydney Boys High School
North Sydney Girls High School
Reddam House
Sydney Boys High School
Sydney Girls High School
Sydney Grammar School
St Aloysius' College

Learning Profile

See exactly where you stand.

Every answer updates your radar and topic progress — so you know what to revise next.

Mathematics Advanced

Apprentice
1.87
Calculus
Disciple
2.40
Exponentials and Logarithms
Apprentice
1.80
Financial Mathematics
Novice
0.40
Functions
Disciple
2.10
Statistical Analysis
Apprentice
1.50
Trigonometry
Master
3.00

A learning profile to indicate your strengths and weaknesses split by topics and subtopics.

Ask Censai

Instant help, whenever you get stuck.

Full worked solution and marking criteria for each question.

Select a topic
Extension 1
Level 2

Question

The diagram below shows the graph of y=ln⁡xy=\ln x.
The shaded region, bounded by y=ln⁡xy=\ln x, the line x=4x=4 and the xx-axis, is rotated 360∘360^{\circ} about the yy-axis to form a solid.
Find the volume of the solid, leaving it in simplest exact form.
Question diagram
Worked solution
Step 1: Express the curve as x=eyx=e^{y}. The shaded region corresponds to 0≤y≤ln⁡40\le y\le\ln 4, with outer radius 44 and inner radius eye^{y} (washer method about the yy-axis).
V=π∫0ln⁡4(42−(ey)2) dyV=\pi\displaystyle\int_{0}^{\ln 4}\bigl(4^{2}-(e^{y})^{2}\bigr)\,dy.
V=π∫0ln⁡4(16−e2y) dyV=\pi\displaystyle\int_{0}^{\ln 4}(16-e^{2y})\,dy.
Step 2: Integrate.
V=π[16y−e2y2]0ln⁡4V=\pi\left[16y-\dfrac{e^{2y}}{2}\right]_{0}^{\ln 4}.
V=π((16ln⁡4−e2ln⁡42)−(0−12))V=\pi\left(\left(16\ln 4-\dfrac{e^{2\ln 4}}{2}\right)-\left(0-\dfrac{1}{2}\right)\right).
V=π(16ln⁡4−162+12)V=\pi\left(16\ln 4-\dfrac{16}{2}+\dfrac{1}{2}\right).
V=π(16ln⁡4−8+12)V=\pi\left(16\ln 4-8+\dfrac{1}{2}\right).
V=16πln⁡4−15π2V=16\pi\ln 4-\dfrac{15\pi}{2}.
Answer: 16πln⁡4−15π216\pi\ln 4-\dfrac{15\pi}{2}
HSC marking criteria

Total: 3 marks

  • 1 mark — Sets up V=π∫0ln⁡4(16−e2y) dyV=\pi\displaystyle\int_{0}^{\ln 4}(16-e^{2y})\,dy (or an equivalent shell integral).
  • 1 mark — Integrates correctly (including the lower-limit contribution +12+\frac{1}{2}).
  • 1 mark — Obtains 16πln⁡4−15π216\pi\ln 4-\dfrac{15\pi}{2}.
Ask Censai

Every subject broken down into topics

Calculus

Vectors

Proof

Statistical Analysis

Trigonometric Functions

Functions

Combinatorics

Calculus

Rates of Change - Time

Exponential Growth and Decay

Related Rates of Change

Further Calculus Skills

Applications of Calculus - Further Area and Volume

Applications of Calculus - Differential Equations

Extension 1

Student Reviews

“Censai showed me exactly where I was weak and gave me the right practice questions to build my foundations. For topics I was confident in, Censai learned this quickly and showed me questions that matched the level of difficulty I needed for my school trials.”
Kevin H.
James Ruse Agricultural High School
“Too often when doing past papers, the questions are too hard or too easy. Censai took the guess work out of exam preparation by learning exactly what level questions I needed to be practising for each topic. The more questions I did with Censai, the better the learning alogrithm.”
Eric Z.
North Sydney Boys High School
“It felt like I had a personal tutor 24/7 and I could get help exactly when I needed it. Other AI tools only provided generic answers, Censai is NESA aligned and always guided me to the answer rather than giving me the complete solution.”
Emma C.
Girraween High School
“This was all I needed to study for my school exams. Tutoring was so expensive and I felt like it was never tailored to my learning speed. Censai persoanlised my learning journey and filled any gaps I had from school.”
Ivy L.
North Sydney Girls High School
“So glad I found Censai before my HSC exams! The more I practised with Censai, the more confident I got with each topic. Being able to see my learning profile progress was great motivation and it honestly felt like a game and I was unlocking all the levels of each topic.”
Edmond F.
Sydney Technical High School
“Learning maths with Censai made me realise what was holding me back at school was not being able to ask questions when I needed to. As Censai learned more about me, the better it got at tailoring my lessons and really felt like I had a private tutor at home.”
Chelsie C.
Baulkham Hills High School

Ready to learn with Censai?

Try Censai

Frequently asked questions

Only HSC Mathematics is available for now. We are working on adding more subjects soon.

Ready to study smarter?

Start a free study session in seconds - quick signup.

Try CensaiSee pricing