LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
Three practice problems in the lecture use one idea: a derivative undoes an integral, and a rate of change needs the chain rule when the quantity you care about is not the one that was given. The derivative of the integral of sin t from 0 to x squared is 2 times x sin(x squared). The area between y = x squared and y = 4 is 32/3. A circular spill whose radius grows at 0.5 meters per minute has area growing at 10 pi square meters per minute when the radius is 10 meters.
Why it matters
The fundamental theorem of calculus says that the integral of a rate from a to b equals the antiderivative at b minus the antiderivative at a. Differentiate both sides and the antiderivative's derivative returns the original function. When a limit of integration is itself a function of x, the chain rule multiplies by that limit's derivative. Area between curves is the same theorem with a difference of functions as the integrand. A related-rate problem is the chain rule without an integral on the page. The lecture treats the three as one sitting, and records that a practice rank rose by one point with accuracy still at 100 percent on that site's board. The rank is not the result. The three closed forms are.
The idea in plain words
The first problem defines F as the area under sin t from a lower bound of 0 up to an upper bound of x squared. Differentiating under the theorem, the upper limit contributes sin of (x squared) and the lower limit contributes sin of 0, which is 0. The upper limit is not x. Its derivative is 2 times x, so the product rule for a variable limit, the chain rule, multiplies: F prime of x equals 2 times x sin(x squared). The lecture submitted that expression and matched an explanation that takes the same two steps. The second problem asks for the area trapped by the parabola y = x squared and the line y = 4. They meet where x squared equals 4, so x is 2 and x is minus 2. On that interval the line is above the parabola, so the absolute value can be dropped and the integrand is 4 minus x squared. Integrate from minus 2 to 2. The antiderivative is 4 times x minus x cubed over 3. At the right endpoint, 4 times 2 minus 8/3. At the left, 4 times minus 2 minus (minus 8/3). Subtracting produces 16 minus 16/3, which is 48/3 minus 16/3, or 32/3. The lecture forgot to write "square" units on the tablet and the grader still accepted 32/3. Symmetry would let you double the integral from 0 to 2. The full integral is what was written out.
How to use it
For an integral with a moving limit, write the integrand at the limit, then multiply by the derivative of the limit. Check a constant limit separately. Here the lower limit 0 adds nothing. For an area, solve the intersection first, decide which function is on top inside that interval, and integrate the difference. Do not integrate from minus 2 to 2 a difference that changes sign, or the absolute value you dropped will have been a mistake. For a related rate, write the geometric formula, differentiate in time, and only then substitute the instant you were asked about. Substituting r = 10 before differentiating loses the factor 2 r and answers a different question.
The lecture remarks that a random version of the spill, with a joint density on the rates, would look more like an interview problem. The deterministic version is the piece to have first. Without dA/dt in terms of dr/dt, there is nothing for a density to average.
What usually goes wrong
Forgetting the 2 times x and reporting sin of x squared. That is the integrand at the upper limit, not the derivative of the integral. Integrating 4 minus x squared from 0 to 2 and forgetting to double, which yields half of 32/3 if the algebra is otherwise right. On the spill, dropping the 2 in the derivative of r squared, or multiplying 10 by 0.5 and forgetting the 2, both miss 10 pi. Units catch some of those. Square meters per minute is an area rate. Meters per minute is the given radius rate. A number with the wrong unit is the wrong number even when the digits look familiar.
Questions people ask
What is F prime for the sine integral?
2 times x times the sine of x squared. The sine of the upper limit comes from the fundamental theorem. The 2 times x is the derivative of x squared.
How did 32/3 appear?
The antiderivative of 4 minus x squared, evaluated from minus 2 to 2, is 16 minus 16/3. That difference is 32/3. The curves meet at x = 2 and x = minus 2.
What instant does 10 pi describe?
The instant when the radius is 10 meters, given that the radius is growing at half a meter per minute. It is not the area, and it is not the rate at a different radius.