Finding Areas
● Can find the area under a curve by integrating
● The same idea can be used to find the area between
a curve and the y-axis
● Instead of integrating with respect to x, we are
integrating with respect to y
● Therefore we use y-values as the limits and we must
express the expression to be integrated in terms of y
Area Between a Line and a Curve
● When finding the area between a line and a curve, we can subtract the
area underneath the curve from the area underneath the line
● Can integrate to find the area under the curve and can use the triangle
area formula to find the area under the line
● Can also integrate both expressions:
● The above approach can also be used to find the area between two curves
Integration by Substitution
● Chain rule allows differentiation of a function of x by making a substitution
with another variable, u
● However, integration is not as easy – we are integrating with respect to x ,
not with respect to u
● Therefore we must change the dx in the equation to a du by multiplying by
dx/du
● This means we divide the function by the derivative of u and then proceed
as normal
● This can often make problems easier, as cancelling can occur
Partial Fractions in Integration
● Often trying to integrate fractions with quadratics or cubics as the
denominator can be exceptionally difficult
● Instead of doing it all at once, split the fraction into partial fractions
● This means each fraction has a simple denominator (e.g x + 1 ) which can
often be integrated using the standard log integrals
Integration by Parts
● Yet another technique for difficult integration
● Can be used to integrate the product of two simpler functions
● This makes it useful in cases where substitution will not help
● The below formula is the formula for integrating by parts – usually you set u
as whichever function is easiest to differentiate
Integrals of Exponential and Logarithmic Functions
Integrals of Trigonometric Functions