8Unit 8: Applications of Integration

The average value of a function over an interval is the constant height that would enclose the same area — you total the function (integrate) and spread it evenly across the width.

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Integration walks the motion chain backwards: integrate acceleration to recover velocity, integrate velocity to recover position.

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Any “rate in / rate out” story is an accumulation: integrate the net rate to get the net change, and add it to a starting amount to get the total present at a later time.

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Between two curves, each thin vertical strip has height (top − bottom).

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When curves are easier to read as $x$ in terms of $y$ (they open sideways), slice with horizontal strips instead: each has width (right − left) and you integrate over $y$.

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If two curves cross several times, which one is on top switches.

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Build a solid from a base region by standing a known shape on each thin slice.

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Same slice-and-add idea, but the standing shape is a triangle or semicircle.

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Revolve a region against its bounding axis and each slice becomes a solid disc.

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Revolving about a line that is not the region’s own axis just shifts the radius: measure the distance from the curve to that line.

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If the region does not touch the axis, each slice is a washer — a disc with a hole.

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Combine the two adjustments: it is a washer (hole) AND the axis is shifted.

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To measure the true length of a curve, add up tiny hypotenuses: each small piece has horizontal run $dx$ and vertical rise $f'(x)\,dx$, so its length is $\sqrt{1+[f'(x)]^2}\,dx$.

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