← Further Maths syllabus

SingaporeFurther Maths

Further Calculus

5 dot points across 5 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do we compute the length of a curve and the area of a surface of revolution by integration?

Calculate the arc length of a curve and the area of a surface of revolution for curves given in Cartesian or parametric form
A focused answer to the H2 Further Mathematics outcome on arc length and surfaces of revolution. The arc-length integral in Cartesian and parametric form, the surface-area-of-revolution formula, and worked applications.
11 min answer →

How do trigonometric and hyperbolic substitutions and standard inverse-function integrals extend our integration toolkit?

Integrate using trigonometric and hyperbolic substitutions and recognise standard integrals giving inverse trigonometric and logarithmic forms
A focused answer to the H2 Further Mathematics outcome on further integration. Standard integrals giving inverse sine and inverse tangent, completing the square, trigonometric and hyperbolic substitutions, and integrals leading to logarithms.
11 min answer →

What is an improper integral, and how do we decide whether one converges and find its value?

Evaluate improper integrals with infinite limits or integrands with a singularity, determining convergence by means of limits
A focused answer to the H2 Further Mathematics outcome on improper integrals. Integrals over an infinite interval, integrands with a vertical asymptote, evaluating them as limits, deciding convergence, and the standard p-integral results.
10 min answer →

How do we derive a Maclaurin series, including by repeated implicit differentiation, and use it for limits and approximations?

Derive Maclaurin series including by repeated implicit differentiation and use series to evaluate limits and approximations
A focused answer to the H2 Further Mathematics outcome on Maclaurin series. The general formula, deriving series by repeated and implicit differentiation, combining standard expansions, and using series to evaluate limits and small-value approximations.
11 min answer →

How do we derive and use a reduction formula to integrate a family of integrals indexed by an integer?

Derive reduction formulae using integration by parts and apply them to evaluate families of integrals
A focused answer to the H2 Further Mathematics outcome on reduction formulae. Deriving a recurrence for an integral by integration by parts, applying it repeatedly down to a base case, and standard reduction formulae for powers of sine and cosine.
11 min answer →