Modulus Functions - Solving Equations Modulus graphs and equations Two non-parallel straight-line graphs would intersect once If modulus involved there could be more than one intersection Deducing ...
Modulus Functions - Sketching Graphs Modulus functions The modulus function makes any ‘input’ positive |x| = x if x ≥ 0 |f(x)| = f(x) if f(x) ≥ 0 |x| = -x if x < 0 |f(x)| = -f(x) if f(...
Modelling with Volumes of Revolution What is meant by modelling with volumes of revolution? Many every day objects such as buckets, beakers, vases and lamp shades can be modelled as a solid of rev...
Adding and Subtracting Volumes Why might I need to add or subtract volumes of revolution? As with the area between a curve and a line or the area between 2 curves, a required volume may be created...
Volumes of revolution around the x-axis What is a volume of revolution around the x-axis? A volume of revolution is the volume of this solid formed How do I find the volume of revolution around the...
Area between 2 curves What is the area between two curves? Ensure you are familiar with … Area under a curve Area between a line and a curve In general find the definite integral of “upper curve” –...
Area between a curve and a line What is the area between a curve and a straight line? This is the area enclosed by a curve and a line between their intersections The intersections may need to be fo...
Area Under a Curve What does area under a curve mean? The phrase “area under a curve” refers to the area bounded by … ... the x-axis … the graph of y = f(x) … the vertical line x = a … the vertical...
Improper Integrals What are improper integrals? An improper integral is a definite integral where one or both of the limits is either: Positive or minus infinity A point where the function is undef...
