The washer method finds the volume of a solid of revolution that has a hole. Each slice is a washer: a disk with outer radius R and a hole of inner radius r. The volume is π times the integral of R(x)² − r(x)², where R reaches the far edge of the region and r reaches the near edge.
The disk method finds the volume of a solid made by spinning a region around an axis. Each slice perpendicular to the axis is a disk with radius R(x), so the volume is π times the integral of R(x)² from a to b. The radius is the distance from the axis of rotation to the edge of the region.
To find volumes by cross sections, slice the solid perpendicular to an axis, find the area A(x) of a typical slice, and integrate A(x) from a to b. For squares A = s², for semicircles with diameter s A = πs²/8, and for equilateral triangles A = (√3/4)s².
The area between curves y = f(x) and y = g(x) from x = a to x = b is the integral of f(x) − g(x), where f is the top curve and g is the bottom curve. The limits are where the curves meet. If they cross in between, split the integral there so each piece is top minus bottom.
Integrals that give inverse trig functions have the square root of a² − u² or the sum a² + u² in the denominator and a constant on top. The integral of 1/√(a² − u²) is arcsin(u/a) + C, and the integral of 1/(a² + u²) is (1/a) arctan(u/a) + C. A third rule gives arcsecant.
Integrals that give natural logarithms have a numerator that is the derivative of the denominator, so the integral of g′(x)/g(x) is ln|g(x)| + C. The simplest case is the integral of 1/x, which is ln|x| + C. The same rule gives the integrals of tan x, cot x, sec x and csc x.
To do u-substitution with definite integrals, change the limits along with the variable. If u = g(x), the new limits are g(a) and g(b). Then evaluate the u-integral with those limits. You never have to substitute back.
U-substitution is a way to find integrals by changing the variable. Let u be an inside function g(x) whose derivative also appears in the integral. Replace g′(x) dx with du, integrate in terms of u, then put g(x) back in for u. It undoes the chain rule.
Displacement is the net change in position, the integral of v(t) from a to b. Total distance counts all motion as positive, the integral of |v(t)|. To find total distance, find where the velocity changes sign, integrate v on each piece, and add the absolute values of the results.
The average value of a function f on an interval [a, b] is the height of the rectangle that has the same area as the region under f. To find it, integrate f over the interval and divide by the interval’s length.