First Fundamental Theorem

MIT 18.01 Single Variable Calculus, lecture 17, explained in plain English

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Part 1 · 0:00–10:00

This part introduces the fundamental theorem of calculus, which says you can find the area under a curve by finding an antiderivative and subtracting its values at two points.

Key ideas

If F' = f, meaning F is an antiderivative of f, then the integral of f from a to b equals F(b) minus F(a). This is the whole theorem. 1:14
The same idea can be written in a shorter notation: put the two endpoints on the bracket and subtract the value at the bottom from the value at the top. 2:35
For the integral of x squared from 0 to b, the antiderivative is x cubed over 3. Plugging in b and 0 and subtracting gives b cubed over 3 in a few seconds, instead of the long method from last time. 4:04
For the area under one hump of sin x from 0 to pi, the antiderivative is negative cos x. Plugging in pi and 0 and subtracting gives an area of 2. 7:16
The method is fast, but the speaker says it is still important to understand the longer method from last time, because the connection between the two ideas matters later. 6:00

Words to know

antiderivative: A function whose derivative gives back the function you started with. If F' = f, then F is an antiderivative of f.
derivative: A measure of how fast a function is changing at each point.
indefinite integral: An integral with no starting and ending points. Its answer is a function, usually with a constant added.
Riemann sum: The slow method of finding area by cutting it into many thin rectangles and adding them up. It was the method used in the previous lesson.

Example

Think of an odometer in a car, which shows the total miles driven. To find how far you drove on a trip, you read it at the end, read it at the start, and subtract. The fundamental theorem works the same way: find the running total (the antiderivative), then subtract its value at the two ends. (This comparison is my own everyday analogy, not from the lecture.)

3 checked practice questions for this part are in the app.

Part 2 · 10:00–20:00

This part explains the Fundamental Theorem using a car trip, then extends it so that negative values and areas below the axis count as negative.

Key ideas

Position is x(t), and its rate of change x'(t) is speed. The integral of velocity from a to b equals the change in position, x(b) minus x(a). 11:04
A Riemann sum imagines checking the speedometer every second. Speed times one second gives the distance for that second, and adding these up gives a close estimate of the total distance. 13:03
The sum is only an approximation. The integral gives the exact distance, because the fundamental theorem says it equals x(b) minus x(a). 14:33
Movement in the negative direction counts as negative. A round trip that ends where it started has a total change of zero, because the forward and backward parts cancel. 16:22
The definite integral is the area above the x-axis minus the area below it. The earlier idea that it is just the area under the curve was incomplete. 19:19

Words to know

Integral: A way of adding up many small pieces to get a total, such as adding up all the tiny bits of distance traveled to get the whole trip.
Riemann sum: An estimate of a total made by cutting something into small time or width pieces, multiplying each piece's size by its value, and adding them all up.
Odometer: The part of a car that counts the total distance driven, like a mileage counter.
Definite integral: An integral between two set points, such as from time a to time b, that gives one number for that stretch.

Example

Think of a car that drives 10 miles to a store and then drives 10 miles back home. The speedometer always shows a positive speed, but the total change in position from start to finish is zero, because the trip out and the trip back cancel each other. The odometer, however, would show 20 miles of total distance.

3 checked practice questions for this part are in the app.

Part 3 · 20:00–30:00

This part explains several basic rules for definite integrals and why they are set up the way they are.

Key ideas

A definite integral measures net distance, which means forward and backward movement cancel out. Total distance would need the absolute value of the velocity, which the definite integral does not use. 21:00
An integral of a sum equals the sum of the separate integrals, and a constant can be pulled out in front. The constant must not depend on x. 23:48
Integrating from a to b and then from b to c gives the same result as integrating from a to c directly. Integrating from a to a gives zero, because there is no width to cover. 25:37
Swapping the start and end points of an integral flips its sign. This is a convention, a rule we agree to use, and it keeps the rules consistent with the fundamental theorem of calculus. 26:46

Words to know

definite integral: A number that tells you the total effect of a function between two points a and b, such as the net change in position.
net distance: The overall distance from start to finish, where movement in the opposite direction takes away from the total.
antiderivative: A function whose derivative is the given function. The lesson uses capital F for the antiderivative of little f.
convention: A rule people agree to follow so that things stay consistent, rather than something proven.

Example

Imagine you walk 5 km east and then 2 km west. Your net distance is 3 km east, but the total distance you walked is 7 km. The definite integral gives the 3 km, because it keeps the direction of each step.

3 checked practice questions for this part are in the app.

Part 4 · 30:00–40:00

This part teaches two shortcuts for working with integrals: the estimation rule for comparing two integrals, and how to change the limits when you substitute a new variable.

Key ideas

Estimation says that if one function is less than or equal to another, then the area under the smaller one is also less than or equal to the area under the bigger one. Think of it as: if one runner is always behind the other, they cover less ground. 30:00
The lower limit a must be smaller than the upper limit b for this rule to work. Swapping the limits flips the sign of the integral. 30:40
You can start with a simple inequality, such as e^x is at least 1 when x is 0 or more, and integrate both sides to get a new inequality. This gives a useful polynomial that approximates e^b. 32:11
Those results only hold when b is 0 or more, because the starting inequality only holds for x that is 0 or more. Using a negative b would give a false conclusion. 34:02
When you change variables and put in new limits, each new limit is found by plugging the old limit into the substitution: u1 = u(x1) and u2 = u(x2). Be careful: this only works if the derivative u' does not change sign. 38:12

Words to know

integral: The total amount added up under a curve between two points, like the area under a graph.
antiderivative: A function whose derivative gives back the function you started with. You use it to evaluate an integral.
substitution: A method of swapping a messy part of an integral for a simpler new variable, so the integral is easier to work out.
estimation: A rule that lets you compare two integrals, using the fact that a smaller function gives a smaller total area.

Example

Imagine two people walking for the same amount of time, where one always walks slower than the other. The slower walker covers less distance, just as a smaller function always gives a smaller integral over the same interval. That is the idea behind estimation.

3 checked practice questions for this part are in the app.

Part 5 · 40:00–48:13

This part shows how to use substitution on a definite integral by changing the limits too, and why a substitution can fail when the function changes direction.

Key ideas

Substitution still works for a definite integral, but you must also handle the limits, the start and end points of the integration. 40:38
The limits belong to the original variable, so you must convert them into values of the new variable u by plugging them into u's formula. 42:33
The most common mistake is to skip this conversion and plug the original x limits into the u answer. The u limits are the ones that count. 42:57
Substitution can break when the substituted function changes direction, as u = x^2 does on the interval from -1 to 1. Then the answer comes out wrong, such as 0 for a positive area. 45:16
To avoid this, split the integral into pieces where the function only moves one way, or pick a substitution where the inverse is well defined. 48:11

Words to know

limits: The start and end values of the integral, showing where to begin and stop adding up.
substitution: A method that swaps a messy piece of an integral for a simpler variable, u, to make it easier to solve.
change of variables: Another name for substitution: replacing one variable with another to rewrite the problem.
inverse: The reverse of a formula, like finding x when you know u, so you can go back and forth between them.

Example

Imagine converting a race course from miles to kilometers. The start and finish markers must be converted too, or you will measure the wrong stretch of road. In the same way, when you change variables in an integral, the limits must be converted into the new variable.

3 checked practice questions for this part are in the app.

← Lecture 18: Definite Integrals

Notes written by AI (Claude) from the lecture's official transcript, so check anything important against the lecture. MIT OpenCourseWare · CC BY-NC-SA 4.0 · not affiliated with or endorsed by MIT. These notes are shared under the same license.

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