Practice question · Sort into groups
A curve is integrated from left to right. Sort each portion by how it contributes to the definite integral.
Groups: Adds a positive amount · Subtracts · Contributes nothing
- A deep trough well below the axis
- A stretch where the curve dips below the axis
- A stretch where the curve sits above the axis
- A stretch where the curve lies exactly on the axis
Hints
- The height f(x) enters each rectangle's area with its own sign.
- A rectangle of zero height has zero area no matter how wide it is.
Show the answer
Adds a positive amount: A stretch where the curve sits above the axis
Subtracts: A stretch where the curve dips below the axis, A deep trough well below the axis
Contributes nothing: A stretch where the curve lies exactly on the axis
Why
The sign of f decides the sign of every rectangle in the sum, so dips below the axis subtract. This is why an odd function integrated over a symmetric interval returns zero even though it plainly encloses area.
Practise The Definite Integral and Area
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