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The graph of g consists of two Ooctags-butosave#C straight lines and semicircle_ Use it to evaluate each integral,Y = gx)g(x) dx(b)K 96) dx9u ) &...

Question

The graph of g consists of two Ooctags-butosave#C straight lines and semicircle_ Use it to evaluate each integral,Y = gx)g(x) dx(b)K 96) dx9u ) &

The graph of g consists of two Ooctags-butosave#C straight lines and semicircle_ Use it to evaluate each integral, Y = gx) g(x) dx (b) K 96) dx 9u ) &



Answers

The graph of $ g $ consists of two straight lines and a semicircle. Use it to evaluate each integral.

(a) $ \displaystyle \int^2_0 g(x) \, dx $
(b) $ \displaystyle \int^6_2 g(x) \, dx $
(c) $ \displaystyle \int^7_0 g(x) \, dx $

The integral from 0 to 2 of G of X. T X is equal to four. The next one leap of the native to pie and for party, we have nine over to bias to pie.

The graph of a function G consists of two straight lines and a semi circle as we see in the following graph. With that information, we want to evil wait the integral in Part eight inter off from 0 to 2 off G of X differential vaccine party the interval off key between two and six. Any perceived in jail of G between zero and seven. So we're going to use the information about what kind of graph we have in each part. Two calculated into growth in a very straightforward way. Simple A. Yeah, We're going to talk about the interval from 0 to 2 off G of X and from 0 to 2, we have the first straight line. We have the other straight line on the interval. Serious. 67 semi circle is strong between two and six as we see here. And thes thread line here on the interval zero to passes through the 0.4 here and through the point to Syria. So we know that this interval here, who represents the area hunter the graph that is under the straight line and above the access Okay, No, within mhm over. If you want interval. Mhm. See her too. Eso this integral corresponds to the area off a writer angle, but is exactly that to the area off the right triangle with furnaces. But 00 here. Yeah, 20 here. Okay. Oh, and Sierra four here is the writer angle and the area we help. You know how to calculate it directly. The area is cool one half times the height of the triangle. That is four units, as we see here. 1234 units times to base of the triangle, which is two units. We can't allow the twos, and we get four. So integral from 0 to 2 of g is four. Okay, because we, uh so that the response to the area of a right triangle, which can be calculated directly. My party, We're going to calculate the interval from C 2 to 6 of she. And over this interval, the function is the semi circle that is completely under the X axis. So the semi circle Oh, has a center. Okay. Thes point here is the center of the circle, and the center then is for zero. And the radius of the A circle is too two unit radius to Uh huh So we can write the equation in the following way. So any point X Y is onto circle ive, and only if the square root off that is the distance off the point X y to the point. For Syria, there is the square root of X minus four square. That's why minus your square is equal to the radius. That is to this equation is equivalent to X minus four square plus y squared equals four, which is equivalent to for square equals four minus explains for square thesis four minus X square minus eight X plus 16 that is four minus X square plus eight x minus 16 and that is negative. X square plus a X minus 12. And that is why square. So it means that we if we consider the complete circle, we don't have function because any point to have two images, that's not a function. But if we take only a semi circle like this like this in this case, we will have a function that it's the portion of the function or the peace of the function that is over the interval to six on the the semi circle, which is below the X axis and because it's below the X axis is always negative. So we know that that's corresponds to the value of why which we take from this equation. By taking the negative square root off this expression. So so the semi circle used in G is why equals negative square root off negative square root of X Negative X square Sorry plus X eight X managed 12. This is the equation because it's the, uh, the result off solving for why the equation of the circle And considering that D square it is got to be taken with negative sign because graph off the semicircle is completely below the y axis. And so, yeah, this will be the equation, but we don't have to make the calculations that is the integral. Okay, from 2 to 6 of g of x, it is true that is equal to negative interval from 2 to 6 of the square root off negative X square plus eight X minus 12. And this will be an admir to solve these mm interval here, but it is not necessary because we use the following recently. We know that, uh D area off or you enclosed by the graph of the function over to six. That is the seventh circle, the X axis. And within 26 Izzy half off the area of the Circle Circle, but with negative sign because the integral is gonna be negative because the function is negative. So we can say here this way, since G of X is the salary co 20 for all X over the interval to six. Then we know that the interval from 2 to 6 of key is a negative number or zero. And we know that the magnitude of this integral Yeah, okay, if we take the absolute value that is half off the area off the entire circle. Okay. As we saw here, we put the positive part of the circle. We will get the area of the whole circle, but in this case, we're gonna have the area off half the circle. But the integral itself is going to be negative because the function is called. It's negative. Overall, the interval to six. That's very important to notice, because you can think of the area. But it's important to notice what sign will be. Will have the integral when we solve directly the internal and not, uh, thinking about the area only. So in this case, we know that the area of the circle yeah, is pie greatest square. That is a spy times the radius of the circle when always to be so here on the graph. So it's too square, and that is 45 then the half half of the area off the circle, which is in fact, the area off, uh, semi circle, sir, my circle is four by over to that is to buy. Then, considering that the injury elicit negative number, the interval from 2 to 6 of G is equal to ah, negative to pa, that is is negative. But the magnitude the number is half off the area of the circle, which is two pi. So we have already calculated this party. Now we go to RC, where we want to calculate the interval from 0 to 7 of tea off key accent and that we know is equal to the integral. With soda, Graf again, we know it's equal to the sum of the integral from 0 to 2, plus interest for 2 to 6 and from 6 to 7. So we know the central is equal to the sum here, Um, a se linearity of the integral. And we have calculated in part a uh, this one and part B, this one. So we get to calculate now this one. And again we see that the area under the graph off if, as in the first part, for the first straight line for the second straight line, the area here is the area of a rectangle. In fact, it's straight a right triangle. I meant a triangle. Mhm area of the right triangle with Virtus is 6070 and 71 The result of the internal is equal to the area with sign included because the functions positive same thing happened for the first red line. The functions positives of the area. It's positive, as does the assist the into law. So we can say that the integral from 6 to 7 of key is equal to the area off the right triangle. Okay. With their diseases. Mm. 60 70 and 71 Uh huh. Uh huh. And the height and base of thes strangle is one and one that is is measure Here is one unit in this measure. Here is one unit, so okay, is integral is equal to one half times one times one that is one half. And oh, So the interval from 0 to 7 off she, as we said above is the some of these three intervals. The first one is in part A is for plus the second internal in party was negative to buy And the interesting Parsi Ah, this one here is one half So yet four plus one half is nine half minus 25 So the integral from 0 to 7 of g of x differential of X is nine half minus two pi. So this is the integral in part definite integral in Parsi. So, as you can see, we can use the idea off area under photograph to calculate rapidly the inter growth. But we gotta be careful about science if the function is completely under or above, the X axis is going to define the sign of the definite integral and we can associate the magnitude with some areas. And with that, we calculate rapidly the values of the injury rules. Yeah,

In this problem, we have to analyze a graph geometrically in order to calculate various integral on the graph. Now, this is a very important skill, especially in math and in calculus is understanding what you're doing geometrically looking at a graph and understanding what you're analyzing and why you're doing it. So the first interval we have is the integral from 0 to 2 ffx in D. X. So the way that I would go about you, um, solving all these inter girls is looking at the interval that will tell you where you're looking at on the graph. So this integral is equivalent to the area of lower quarter of the circle with the radius too. So we would take the integral would be equivalent to negative 1/4 times pi r squared. And when we simplify that we would get that are integral is equivalent to negative pie. Now on to part B, we have the integral from 2 to 6 off death affects in D X. This would be equivalent to the area of the triangle over the X axis. Also, something to note about these problems will keep referencing the graph that's going to help you a lot. I've only included it on the top. But when you're going through this problem, I would keep referencing it. So for this integral, that would be equivalent to one half times four times to. And when we simplify that the evaluated integral before under part C, we have the integral from negative 4 to 2 of ffx in D. X. So what would this be equivalent to This would be equivalent to the area of the triangle below the X axis, plus the area of the semi circle. So we would have first the area of the triangle when we would have, or the area of the semi circle it have negative one half times two times one plus one half times pi times two squared. And when we simplify that, we would get negative one plus two pi. Now, the reason why I said you could do this in either order is because you can, um, addition. You could reverse the order that this is in and you'd get the same answer. I want to part D. This one's a little bit more involved, but this is combining everything that we've essentially looked at. So we have the integral from negative 4 to 6 of ffx in D. X, a very useful property of integral XYZ. We can split them up into different intervals and add them all together. So that's what we can do for this. This would be equivalent to the integral from negative 4 to 2 of ffx indie X plus the integral from negative to to to of ffx indie X plus the integral from 2 to 6 and ffx indie X. So then we would add up all of those inter girls we would have negative one half times two times one plus negative one half pi times two squared plus one half times four times to then you would simplify that, um, completely across. And we would get that this original integral is equivalent to three minus two pi. And now for E, we have the integral from negative 4 to 6 of the absolute value of ffx in D x. This would be the some of the absolute value of our answers for B and C. Basically, we just add them together so we would have four plus one plus two pi. We've simplified that to get five plus two pi and finally are integral for F is the integral from negative 4 to 6 of F of X plus two Indy X, so we can separate that to very easily. So we get that this is equivalent to the integral from negative 4 to 6 of ffx indie X plus the anti derivative of two, which is two X evaluated from negative 4 to 6. When we simplify that in this portion, you could do B minus a like how you solve any definite integral we would get that this is equivalent to the integral from negative 4 to 6 of f of x in D X plus 20. So, how we solve this, we would take this first integral three minus two pi which we've found before, and then we'll add 20. So we'll get that. The solution to that integral is 23 minus two pi. So I hope that this problem helped you understand how we can analyze it graft geometrically and use that information to calculate various intervals using the intervals on the graph


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