Problem

Use a calculator and right Riemann sums to approx…

02:51

Problem 57 Hard Difficulty

Use a calculator and right Riemann sums to approximate the area of the given region.
Present your calculations in a table showing the approximations for $n=10,30,60,$ and 80 subintervals. Make a conjecture about the limit of Riemann sums as $n \rightarrow \infty.$
The region bounded by the graph of $f(x)=\frac{1-\cos x}{2}$ and the $x$ -axis on the interval $[-\pi, \pi]$.

Answer

n Right Riemann sum
10 3.14159
30 3.14159
60 3.14159
80 3.14159

Related Courses

Calculus 1 / AB

Calculus 2 / BC

Calculus Early Transcendentals

Chapter 5

Integration

Section 1

Approximating Areas under Curves

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Video Transcript

Okay, So in this question, we are given a function over a specific interval. Eight were adds to calculate the right Riemann. Some given four different numbers of seven rolls. So I've got the answers, written it and table over here on the left. Ah, let question asked for And, um what weekend? How we need to go get a beach value as we first start off by getting Delta X, uh, which is our range divided by our number of seven rolls. And then we need funding, um, our endpoint values for each sub in Rome my okay, are XK values. So X cables, a beginning of the first sub roll, plus k times delta X, the length of each step in the role and then lastly, way to plug in. So we're gonna get expression for that rather than an exact value. We're going to keep ks a, um, variable there, and then we're gonna plug that expression into the function that we were given. Um f so that's FM XK. Ah, and we're gonna sum that up from one to end on end while also multiplying by Delta X. So you do that and you're gonna need your calculator to do that. Um, then you will be able to get the answers on left here in the table and impact be asked us a boat. Um, the approach, the value Reitman some as, and the number of seven year olds purchase infinity. Well, that's even see, um, it's it's not changing. Our values were all 3.14159 approximately. And it actually keeps going because as I'm sure, you guys all realized the opening digits of pi. So, um, as an approaches infinity, um, the answer won't change. The area under curve will continue to be, uh, by

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