Antiderivative Steps: the Reverse Power Rule, + C, and the Total Between Two Times

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Opens on Chapter 20’s example: the hobby rocket’s speed, 30t² + 20t feet a second, from launch to 1.5 seconds.

Rate (a polynomial) one letter, numbers, + − × ÷ ^ ( ), ² ³, e, π, √, ln

From

Starting value (C), if you know it e.g. an odometer reading

Value at the time optional

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Enter your numbers, then press Work it out.

How it works

The reverse power rule undoes the power rule one term at a time: add one to the power, then divide by the new power, so t² becomes t³/3. Check by differentiating: you should get the rate back. Any number C can be added, because a constant has zero slope; C is where the total started (an odometer's reading before the trip). The total between two times is F(b) − F(a), and the C cancels. The picture shows the same total as the area under the rate.

From the book, Chapter 20. F(t) = t³ ÷ 3 gives F(4) − F(0) = 21.333 for the go-kart. The rocket rises 56.25 feet in 1.5 s. The tank takes in 600 gallons. Archimedes’ area is 1/3. Gravity alone gives d = 16t²: 64 feet in 2 s, or 84 if thrown down at 10 ft/s. The odometer reads 41,249 after 3.5 hours. Every chapter and its calculator →

Numbers can be typed as 1,450, $38, 0.40, 12%, 1/6 or 2 3/8. Functions can be typed as 16t², 6 + 48t − 16t^2, 1000e^(0.05t) or √x; answers are rounded half up, the way the book rounds. Everything stays in your browser: no sign-up, nothing stored or sent.