Unit 1: The Definite Integral and the Fundamental Theorem of Calculus
A fast review of the definite integral and the Fundamental Theorem, sharpened for everything ahead.
6 topics · 24 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Riemann Sums and the Definite Integral
- How it worksHow does a Riemann sum turn a pile of rectangle areas into the definite integral as n grows without bound?
- Worked exampleFind the right-endpoint Riemann sum for f(x) = x2 on [0, 2] using 4 rectangles.
- Common mistakeWhy isn't a Riemann sum just a rough guess that never actually equals the true area?
- Going furtherWhat happens to a Riemann sum's accuracy when you switch from right endpoints to midpoints?
Your turn1 of 2
Find the right-endpoint Riemann sum for f(x) = x2 on [0, 3] using n equal-width rectangles, as a function of n.Properties of Definite Integrals
- How it worksHow do the additivity and constant-multiple rules let a hard integral split into easier pieces?
- Worked exampleFind the integral from 0 to 5 of f(x) dx given the integral from 0 to 3 of f is 7 and from 3 to 5 of f is 4.
- Common mistakeWhy doesn't integrating over a longer interval always produce a bigger number?
Your turn1 of 2
Given that the integral from 0 to 3 of f(x) dx equals 7 and the integral from 3 to 8 of f(x) dx equals -1, find the integral from 0 to 8 of f(x) dx.The Fundamental Theorem, Part 1: The Accumulation Function
- How it worksHow does differentiating an accumulation function just hand you back the original integrand?
- Common mistakeWhy isn't finding the derivative of an accumulation function a job that needs a second integral?
- Worked exampleFind g'(x) for g(x) equal to the integral from 0 to x of √(t2 + 1) dt.
- Worked exampleFind h'(2) for h(x) equal to the integral from 1 to x2 of cos(t) dt.
Your turn1 of 2
Let g(x) be the integral from 0 to x of √(t2 + 4) dt. Find g'(x).The Fundamental Theorem, Part 2: Evaluating with an Antiderivative
- How it worksHow does the difference F(b) minus F(a) collapse an infinite sum of rectangles into two evaluations?
- Going furtherWhat changes about evaluating an integral when the antiderivative has a removable discontinuity inside the interval?
- Worked exampleFind the integral from 1 to 3 of (2x + 1) dx using an antiderivative.
- Common mistakeWhy doesn't the constant of integration matter when you evaluate a definite integral?
Your turn1 of 2
Find the integral from 1 to 5 of (2x + 5) dx using an antiderivative.The Net Change Theorem
- How it worksHow does the net change theorem turn a velocity function into total displacement over an interval?
- Worked exampleFind the net change in position over [0, 5] for a particle with velocity v(t) = 3t2 - 12t.
- Worked exampleFind the total change in temperature over 4 hours given a rate of 2cos(t) degrees per hour.
- Common mistakeWhy doesn't integrating a velocity function that goes negative give you the total distance traveled?
- Going furtherWhat happens to the net change theorem when the rate function is only piecewise defined?
Your turn1 of 2
Find the net change in position over [0, 5] for a particle with velocity v(t) = 4t2 - 4t.The Average Value of a Function
- How it worksHow does dividing an integral by the length of the interval turn area into an average height?
- Worked exampleFind the average value of f(x) = x2 on the interval [0, 3].
- Common mistakeWhy isn't the average value of a function the same as the average of its highest and lowest values?
- Ties togetherHow does the average value of a function connect to the Mean Value Theorem for derivatives from Calculus I?
Your turn1 of 2
Find the average value of f(x) = 4x2 on the interval [0, 5].