Calc Class: Set Up Right-Hand Riemann Sums, Approximate Integrals, Evaluate Integrals with FTC

We want to find the area under the graph of f(x)=x^2 from x=2 to x=5. We first approximate the area with a right-hand Riemann sum. This is approximated in two ways using "Delta x" = 0.75 and n = 4. Then the limit as n goes to infinity is found to find the exact area. The Fundamental Theorem of Calculus is introduced and used to confirm the answer to equal 39. The sine function is integrated from 0 to pi to get 2. The function f(x)=(3/2)/x^2 is a probability density function for x between 1 and 3. If the velocity is v=f(t)=1-t^2 for t between 0 and 2, then the definite integral is negative and gives the change in position. Finally, if the function is always increasing or always decreasing (monotone function), then the distance between the left-hand sum and right hand sum is the absolute value of f(b)-f(a) times "Delta t".

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⏱️TIMESTAMPS⏱️
(0:00) Today the topic will be definite integrals.
(1:12) Given a continuous function on [a,b].
(2:26) Assume f(x) is non-negative for simplicity.
(3:16) We want to find the area under the curve.
(6:14) Example 1: f(x)=x^2 over [a,b]=[2,5].
(6:53) Steps of Riemann sum approximation (Right-Hand Sum), based on a partition of [a,b] into subintervals of length "delta x" = (b-a)/n.
(16:39) Approximation when n=4.
(20:50) Symbolic right hand sum (involves the sum of the first n integers and the first n perfect squares).
(34:19) Limit of Riemann sum gives the exact area.
(36:34) Confirm the answer with the Fundamental Theorem of Calculus.
(40:01) Check with Mathematica.
(43:57) Example 2: Find the area under the sine function as x goes from 0 to pi.
(47:22) Example 3: Probability density function application.
(54:37) Example 4: Get change in position (displacement) given velocity v=f(t)=1-t^2 for t from 0 to 2 by integrating to get a signed area that ends up negative.
(1:00:28) Distance between left-hand sum and right-hand sum when the function is monotone.
(1:02:34) Homework question (find the sampling rate for the velocity).

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