Integration by Parts Picture

Lesson notes

Overview

Area under one curve relates to area under its partner: ∫u dv = uv − ∫v du. This topic is rated intermediate and is number 172 of 245 in Mathematics. Builds on the basics with relationships, methods and trade-offs.

What you are seeing

The panel above shows a graph of the function. Each slider changes one parameter, so you can see exactly how the curve bends, shifts or stretches. The animation is drawn live from the model, so everything you see responds to the sliders. Press Space to pause, use Restart to replay from the start, and change the speed to slow down quick events.

In depth

Integration by parts, ∫u dv = uv − ∫v du, is the product rule run backwards. In a picture of the curve in the (u, v) plane, the area under one curve and the area under its inverse add up to the whole rectangle uv. Choose u to be a function that gets simpler when differentiated and dv to be something easy to integrate. It solves integrals such as ∫x eˣ dx and ∫ln x dx.

Reading the relationship

The expression "∫ u dv = uv − ∫ v du" is a compact summary of the idea. It uses the symbols u, d, v; name what each one stands for before you start. Notice that it contains an integral, which accumulates a quantity over an interval. Then use the sliders to see how changing one quantity changes the others.

A guided exploration

1. Set every slider to its middle value and note the shape of the curve. 2. Move one slider at a time and describe whether the curve shifts, stretches or turns. 3. Find the value where the behaviour changes sharply, then connect it to the formula.

Try this

Predict the outcome first, then test extreme slider values to find where the pattern changes or breaks.

Common pitfalls

A frequent slip is memorising the rule without the picture. If a result surprises you, sketch the graph or test a simple number by hand before trusting the algebra.

How it connects

It builds on “Area Under a Cosine Hump”, the topic before it in Mathematics. Once you are comfortable, continue with “Antiderivatives and Constants”.

Why it matters

Mathematical ideas like this one are the language that physics, engineering, computing and statistics are written in.