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Unit 8
10–15% of examIn this unit you put integrals to work. You use them to find average values, track how far something moves, total up an amount that changes over time, and measure areas between curves and the volumes of solids. The habit that ties it all together: picture one thin slice, write its area or volume, then add up all the slices with an integral.
Longer videos that cover the whole unit. Good for a first pass or a final review.
The average value of f on [a, b] is the integral of f from a to b divided by (b − a). Picture it as the height of a rectangle over [a, b] with the same area as the region under the curve.
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Integrating velocity over a time interval gives displacement (the change in position), while integrating speed, |v(t)|, gives the total distance traveled. To find a later position, add the displacement to the starting position: x(b) = x(a) + ∫ from a to b of v(t) dt.
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If you know the rate at which something changes, the definite integral of that rate gives the total change over the interval. Add the starting amount to get the final amount, and always give units, such as gallons or people.
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To find the area between two curves, integrate (top curve − bottom curve) with respect to x. The limits are the x-values where the region starts and ends, often where the curves cross.
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When a region is easier to describe from left to right, slice it horizontally and integrate (right curve − left curve) with respect to y. The limits are then y-values, and each curve needs to be written as x in terms of y.
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When two curves cross more than twice, the one on top switches at each crossing. Split the integral at every crossing point so you always subtract bottom from top, then add the pieces; or integrate the absolute value of the difference, which is handy on a calculator.
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Some solids sit on a flat base region, and every slice perpendicular to an axis has the same shape. If each slice is a square with side s(x), its area is s(x)², and the volume is the integral of that area along the axis; for rectangles, multiply the base by the height you're given.
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It's the same slicing idea with different shapes, so you just need the right area formula. A semicircle with diameter s has area (π/8)s², and an equilateral triangle with side s has area (√3/4)s².
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Spinning a region that touches the axis makes a solid whose slices are discs. Each disc has area πr², where r is the distance from the axis to the curve, so the volume is π times the integral of r².
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When a region spins around a line such as y = 3 or x = −1, the radius is the distance from that line to the curve, for example 3 − f(x). Sketch the radius first, then square it inside the integral.
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When the region doesn't touch the axis, each slice is a washer: a disc with a hole in the middle. Its area is π(R² − r²), with R the outer radius and r the inner radius, so you subtract the squares, not the radii.
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This is the washer method again, but both radii are measured from the new line. For example, if the region sits above the line y = −2 and spins around it, a curve y = f(x) is f(x) + 2 away from the axis.
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