Integral as Area

Skill goal: connect the definite integral to signed area.

Observe

The shaded region runs from a (blue handle) to b (teal handle). Region above the x-axis counts positive; region below counts negative. The integral is the net signed area.

Manipulate

Function:

area above axis (positive)area below axis (negative)

Tip: drag the blue handle (a) and the teal handle (b) along the x-axis. Try dragging b to the left of a!

Predict: Before you change it, what do you think will happen? If the blue region and the teal region are the same size, what will the integral be?

See

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∫_a^b f(x) dx (net)

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Above / below

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Explain

Why it works: The definite integral adds up signed area between the curve and the x-axis. Area above the axis adds to the total; area below the axis subtracts from it. Equal regions on opposite sides cancel — that is why an integral can be zero even when the picture is full of shaded region. Reading right-to-left (b < a) flips the sign of the whole integral.

Challenge

1. Switch to f(x) = sin x. Drag a and b to nonzero positions so the net area is 0 (hint: symmetry). Then press Check.

2. Switch to f(x) = x². Set a = −1 and b = 1 (drag both handles). Then press Check.

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