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...
Reverse Chain Rule How do you integrate (ax?+?b)n? The reverse chain rule can be used for integrating functions in the form y = (ax?+?b)n Make sure you are confident using?the chain rule?to?differe...
