Integration
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:
a
○ Area = ∫ (f (x) − g(x))
b
○ f (x) = equation of line, g (x) = equation of curve
● 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
du
dx
● 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
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:
a
○ Area = ∫ (f (x) − g(x))
b
○ f (x) = equation of line, g (x) = equation of curve
● 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
du
dx
● 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
