Summary — What This Chapter Covers
- ∫xn dx = xn+1/(n + 1) + c for n ≠ −1.
- ∫(ax + b)n dx = (ax + b)n+1/[a(n + 1)] + c for n ≠ −1.
- Use a known point on a curve to find the constant of integration.
- Definite integrals give signed area; areas below the x-axis come out negative.
- Simple improper integrals: an infinite limit, or a function undefined at one limit.
- Volume about the x-axis: π∫y2 dx; about the y-axis: π∫x2 dy.
1. Indefinite Integration
Reverse differentiation
Add one to the power, divide by the new power
Rewrite roots and fractions as powers first. For a bracket (ax + b)n, integrate as if the bracket were x and then divide by a, the derivative of the inside. If you are given dy/dx and a point on the curve, integrate and substitute the point to find c.
Rules
∫xn dx = xn+1/(n + 1) + c · ∫(ax + b)n dx = (ax + b)n+1/[a(n + 1)] + c (n ≠ −1)
Worked Example
A curve has dy/dx = √(2x + 1) and passes through (4, 10). Find its equation.
Practice Question 1
Find ∫(3x − 2)4 dx and ∫ 1/(2x + 3)2 dx.
Practice Question 2
Find ∫(6x2 − 4/x3) dx.
2. Definite Integrals and Area
Area
Under a curve, between curves, and to infinity
∫ab f(x) dx = F(b) − F(a). Area between a curve and the x-axis is ∫y dx; if part of the region is below the axis, that part gives a negative value, so find each part separately and add the sizes. Area between two curves is ∫(upper − lower) dx between their intersections. Area against the y-axis is ∫x dy.
An improper integral has an infinite limit, or an integrand that is undefined at a limit. Evaluate with a letter, then see what happens as it tends to the limit: ∫1∞ x−2 dx = [−1/x]1∞ = 0 − (−1) = 1, and ∫01 x−½ dx = [2√x]01 = 2.
Worked Example
Find the area enclosed between y = 4x − x2 and y = x.
Worked Example
Find the area between y = x2 − 4x and the x-axis.
Practice Question 1
Evaluate ∫14(2x + 1/√x) dx.
Practice Question 2
Evaluate ∫1∞ x−3 dx.
3. Volumes of Revolution
Volumes
Rotate through 360°
Rotating y = f(x) between x = a and x = b about the x-axis gives a solid of volume π∫y2 dx. About the y-axis, write x2 in terms of y and use π∫x2 dy with y-limits. If the region is not bounded by the axis, subtract: π∫(youter2 − yinner2) dx.
Volumes
a
b
2
c
d
2
Memorise