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Evaluate the following integrals. State and verify the theorem(s) used, include sketch of the given contour C where relevant:where € is the polygonal path con...

Question

Evaluate the following integrals. State and verify the theorem(s) used, include sketch of the given contour C where relevant:where € is the polygonal path consisting of the line segments from =0 t0 2 = 2 and from 2 = 2 t0 2 = 1+ i.

Evaluate the following integrals. State and verify the theorem(s) used, include sketch of the given contour C where relevant: where € is the polygonal path consisting of the line segments from =0 t0 2 = 2 and from 2 = 2 t0 2 = 1+ i.



Answers

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. Sketch the graph of the integrand and shade the region whose net area you have found. $$\int_{1 / 2}^{2}\left(1-\frac{1}{x^{2}}\right) d x$$

Into a cruel When are why True deal issue number return, as in cathedral, begins from minus one toe bone integral begins from minus K X minus. Boy to peace close to try to Thievery Gate, Michigan would toe integral begins from minus one toe on boy three. Devoid is worried three minus X minus one to east Lost to days A community haunts toe. Wonder. Why does boy to be integral from minus one toe? One tool X knows to three minus. He's applying this used minus one three the which equals toe. Wanted going into, like three to XT plus two line. They were the boy cage No minus gigs. Wine this born or avoid this boy Well ro minus phone. Who? One? Mitchell was true. Mon. Divided by three or four divided by age. Yes, minus two. Cool the boy. This boy minus one, divided by three zero for the wouldn't really eight lost serial bull. He really does blame for it. It goes to one

Problem Number 58. You think the computer and your resistance the integral is equal to five over three square root for three, which approximately equal toe 0.6 or four six, and you might get a different result depend on the CS being used blood approximation must be equal.

So, you know, I think this into in total four. Excuse why? I mean, thanks. Plus, next to the fourth, do you? Why do you want to? You've always before see a closed. Mm. So? Well, Lo Green's theorem tells us, but really this Well, this would be, uh, being drill for some folks from Vector field if, but, uh, its components four. You Why? I mean, along the second component extra before show greens by green theory. I am green means Yeah, um, this is gonna be called to the great all over the interior will see your licks. You know it like that. The interior being dear area off the surface of the curve in close Latiker there are people in this M I'm in would be partial ban respect. Toe on respect, Rex. Miners partial man with respect to y. So, uh, so that if we have ah, you have these partial. These with respect to X is gonna be partial in respect. Works would be three. Next. The superpower minds and impartial thes who respect. Why would be, um no how these journeys more differentiated room myself. This is this papers would give us or extra assert and they differentiating doesn't respect. Why would be We're just you're you do the same. Tokyo is a whole the x t y your friendship over you so we can see here that this to cancel So you're intimating zero Reese era times these, But is the area closed by the curve? So these day area off here So Sierra Times area of interior is equal to zero. So that these this interval where in close blast is gonna be equals zero As long as we have our loss both.

Okay, So you want to solve this in a role? Uh, the double inter goal over our of ah, square root of X Great plus y squared. Uh, d a Where are is this region? Ah, x squared Plus are by security is between one and four. Now, knowing our public ordinance, we know that this is just, um r squared is between four and one or ours between two in one. So in the three d plane, this is just a minute tooty, uh, if I playing this is just the area between these two circles, this red area kind of like a circle with a hole in the middle. And since it's just a circle, there's no other bounds on X. And why we know that this is just goes from, uh, data from 0 to 2 pi. So it's pretty simple. We know that this is just great route of r squared or just are and d A is just already already thing. So we have this in a religious following, so it's just r squared d r D data bringing our sub separating or intervals. We end up with this, um, sorry. R squared er, giving US A. Two pi times are queued over three from 1 to 2 or two pi times 8/3 minus 1/3 or 14 pi over three.


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