doPhysics · Mathematical Tools
Integration
Integration from the beginning. The antiderivative, the integral as the area under a curve, a short table of integrals with substitution, integrals over a body, mean and root mean square values, and separable differential equations.
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Why this chapter
Why this chapter exists
Differentiation takes a quantity and finds its rate of change. Physics just as often gives the rate and asks for the quantity. A speedometer record must become a distance, a changing force must become work, and a rate of decay must become the number of nuclei left. Integration is the tool that runs this journey. It adds very many small pieces, strips under a graph or elements of a body, into one total. Every unit of mechanics, heat, electricity and modern physics uses it.
You will learn
By the end of this chapter you will
- Find an antiderivative by the reverse power rule, and fix its constant from a starting value.
- Explain why the area under a curve is the limit of a sum of thin strips, and find it as \(F(b) - F(a)\).
- Use a short table of integrals, substitution and integration by parts on the integrals of physics.
- Cut a rod, a disc or a sphere into elements, and find its mass, centre of mass or moment of inertia.
- Find the mean and the root mean square value of a changing quantity, and say what 220 V rms means.
- Solve a separable differential equation, and use it for decay, charging and terminal velocity.
Before you begin
Before you begin
You should know what a derivative is and how to differentiate powers, sines, cosines and exponentials. The chapter Differentiation, just before this one, gives all of that. You should be able to read a graph and substitute numbers into a formula. Nothing else is assumed; integration is built here from the start.
Lessons
1
Undoing the Derivative
The antiderivative, the reverse power rule, the constant of integration and the starting value that fixes it, used to find velocity and position from acceleration.
The antiderivativeThe constant is fixed by a starting valueFrom acceleration to velocity to position
2
The Integral Is the Area Under a Curve
The definite integral as the sum of thin strips, why the area undoes the derivative, and how one antiderivative gives the displacement under a velocity graph and the work under a force graph.
The integral is the area under a curveWhy the area undoes the derivativeSigned area and area between curves
3
Standard Integrals and Substitution
The short table of integrals that physics uses, the logarithm where the power rule fails, substitution as the chain rule used in reverse, and integration by parts for a product.
Standard integrals and substitutionSubstitution undoes the chain ruleIntegration by parts for a product
4
Integrating over a Body
Cutting a rod, a disc or a sphere into elements, writing each element's mass as \(dm = \lambda\,dx\), and adding the elements to find a mass, a centre of mass and a moment of inertia.
Integrating over a bodyThe balance point of a rodRings and shells build discs and spheres
5
Average Value and Root Mean Square
The mean of a changing quantity as the height of a rectangle of equal area, why \(\sin^2\) averages to one half, and the root mean square that gives the 220 V of the mains.
Average value and root mean squareThe mean of sine squared is one halfRms of any waveform
6
Separable Differential Equations
Equations that give a rate of change, solved by separating the variables and integrating both sides; the four the book meets are constant acceleration, radioactive decay, a charging capacitor and a body falling against a drag.
Separable differential equationsDecay at a rate proportional to the amountApproach to a final value
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- antiderivative
- A function whose derivative is a given function. \(F\) is an antiderivative of \(f\) when \(dF/dx = f\).
- constant of integration
- The constant \(C\) added to every antiderivative. It is fixed by one known value, such as the value at the start.
- definite integral
- The limit of a sum of thin strips \(f(x)\,\Delta x\) from \(x = a\) to \(x = b\), written \(\int_a^b f(x)\,dx\). It is the signed area under the graph of \(f\).
- differential equation
- An equation that contains a derivative of an unknown function, such as \(dN/dt = -\lambda N\). Solving it finds the function.
- element
- A piece of a body so small that all its mass has, very nearly, one position or one distance from an axis.
- fundamental theorem of calculus
- The result that the area under \(f\) from \(a\) to \(b\) equals \(F(b) - F(a)\) for any antiderivative \(F\) of \(f\). It shows that integration undoes differentiation.
- indefinite integral
- The family of all antiderivatives of a function, written \(\int f(x)\,dx = F(x) + C\).
- initial condition
- The known value of a quantity at the start, such as the position or the velocity at \(t = 0\). It fixes the constant of integration.
- integrand
- The function inside an integral sign, the \(f(x)\) in \(\int f(x)\,dx\).
- integration by parts
- A method for integrating a product, \(\int u\,dv = uv - \int v\,du\). It is the product rule used in reverse.
- limits of integration
- The two values \(a\) and \(b\) between which a definite integral is taken; \(a\) is the lower limit and \(b\) the upper limit.
- linear density
- The mass per unit length of a rod at a point, \(\lambda\), in kg/m. An element of length \(dx\) has mass \(\lambda\,dx\).
- mean value
- The integral of a quantity over an interval divided by the length of the interval. It is the height of a rectangle with the same area.
- root mean square
- The square root of the mean of the square of a quantity. For a sine of peak \(x_0\) it is \(x_0/\sqrt2\).
- separable
- Describes a differential equation whose rate is a product \(f(x)\,g(y)\), so that each variable can be moved to its own side before integrating.
- substitution
- A method of integration that puts a new variable \(u\) for an inner expression and \(du\) for the matching factor. It is the chain rule used in reverse.
- terminal velocity
- The steady velocity of a falling body at which the drag balances the weight, so that the acceleration is zero.
- time constant
- The time \(\tau\) in which a quantity that approaches a final value closes all but \(1/e\), about 37 %, of the gap to it.