Welcome to Further Integration and Kinematics!

In your Standard Level studies, you learned how to find the area under a curve and how to integrate simple polynomials. Now, we are stepping it up for Higher Level! This chapter is all about taking those tools and applying them to more complex functions, 3D shapes (volumes of revolution), and the physics of motion (kinematics). Whether you want to be an engineer, a physicist, or a data analyst, these tools are essential for modeling how the world moves and grows.

1. Further Integration Techniques

At HL, we expand our "integration toolkit" beyond just simple power rules. We now include trigonometric functions and a powerful method called substitution.

New Standard Integrals

You need to be comfortable with these specific results. Don't forget the constant of integration \(C\) for indefinite integrals!

  • The Natural Log Rule: When you integrate \(x^{-1}\) (or \(\frac{1}{x}\)), the power rule fails. Instead: \(\int \frac{1}{x} dx = \ln|x| + C\).
  • Trigonometric Functions:
    • \(\int \cos x dx = \sin x + C\)
    • \(\int \sin x dx = -\cos x + C\) (Watch that negative sign!)
    • \(\int \frac{1}{\cos^2 x} dx = \tan x + C\)
  • Rational Powers: The power rule \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\) now applies to any rational number \(n\) (except \(n = -1\)).

Integration by Substitution

Think of this as the "Reverse Chain Rule." We use it when one part of the function is the derivative of another part. The syllabus focuses on the form: \(\int f(g(x))g'(x) dx\).

Step-by-Step Substitution:

  1. Choose a part of the function to be \(u\) (usually the "inside" function).
  2. Find the derivative \(\frac{du}{dx}\).
  3. Rearrange to replace \(dx\) with an expression involving \(du\).
  4. Substitute everything into the integral so only \(u\) remains.
  5. Integrate with respect to \(u\).
  6. Substitute the original \(x\) expression back in.

Example: To find \(\int 2x \cos(x^2) dx\), let \(u = x^2\). Then \(\frac{du}{dx} = 2x\), so \(du = 2x dx\). The integral becomes \(\int \cos(u) du\), which is \(\sin(u) + C\). Putting \(x\) back in, we get \(\sin(x^2) + C\).

Quick Tip: If you see a function and its derivative multiplied together, substitution is almost always the way to go!

2. Volumes of Revolution

Imagine taking a 2D curve on a graph and spinning it 360 degrees around an axis. You've just created a 3D solid! We use integration to find the volume of that solid.

Rotation about the x-axis

When we rotate a curve \(y = f(x)\) around the x-axis between \(x = a\) and \(x = b\), the formula is:

\(V = \pi \int_{a}^{b} y^2 dx\)

Why \(\pi\) and \(y^2\)? Think of the solid as a stack of tiny circular "slices." Each slice has a radius of \(y\). The area of a circle is \(\pi r^2\), so each slice has an area of \(\pi y^2\). Integration just adds all those slices together!

Rotation about the y-axis

If we rotate the curve around the y-axis between \(y = c\) and \(y = d\), we use:

\(V = \pi \int_{c}^{d} x^2 dy\)

Common Mistake: Students often forget to rearrange their equation. If you are rotating around the y-axis, your integral must be in terms of y. You must change \(y = f(x)\) into \(x = g(y)\) before squaring it!

3. Areas between Curves and the y-axis

While SL focuses on the area between a curve and the x-axis, at HL you may need to find the area between a curve and the y-axis.

The formula is: \(A = \int_{c}^{d} x dy\)

Just like with volumes, you must rearrange your function to get \(x\) by itself before you integrate with respect to \(y\).

4. Kinematics (HL)

Kinematics is the study of motion. We use three main variables: Displacement (\(s\)), Velocity (\(v\)), and Acceleration (\(a\)).

The Calculus Connection

Remember this flow: Displacement \(\to\) Velocity \(\to\) Acceleration.

  • To go "down" the chain (finding rates of change), we differentiate:
    • \(v = \frac{ds}{dt}\)
    • \(a = \frac{dv}{dt} = \frac{d^2s}{dt^2}\)
  • To go "up" the chain (finding totals), we integrate:
    • \(v = \int a dt\)
    • \(s = \int v dt\)

Displacement vs. Total Distance

This is a classic IB exam trap! Don't worry, it's easy once you see the difference.

  • Displacement: Where you are relative to where you started. Found by \(\int_{t_1}^{t_2} v(t) dt\).
  • Total Distance: Every step you took, regardless of direction. Found by \(\int_{t_1}^{t_2} |v(t)| dt\).

Analogy: If you run 100m forward and then 100m back to the start, your displacement is 0m, but your total distance is 200m. On your GDC (Graphic Display Calculator), use the absolute value function to find total distance easily!

Did you know? Acceleration isn't always constant in these problems. You might have acceleration that changes over time, like \(a(t) = \sin(t)\). This is why calculus is so much more powerful than the basic "SUVAT" equations you might use in Physics!

Summary Checklist

Key Takeaways:

  • Always add \(+ C\) for indefinite integrals.
  • Remember that \(\int \frac{1}{x} dx = \ln|x|\).
  • For Volumes of Revolution, don't forget the \(\pi\) outside the integral and to square the function (\(y^2\) or \(x^2\)).
  • When integrating with respect to \(y\), make sure your limits (\(c\) and \(d\)) are y-values.
  • For Kinematics: Integrate velocity for displacement; integrate the absolute value of velocity for total distance.

Note: For further exploration of how integration applies to differential equations, see the chapter on "Differential equations, slope fields and Euler's method (HL)".