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Which of the following areas are equal? Why?

$A=B=C$

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Campbell University

Oregon State University

Harvey Mudd College

Baylor University

all right. We got the question with three graphs and were asked if any of the areas are equal to one another. I'm not gonna copy the graph over, but you can see that with each graph there is arrange Example the first one there's Y e to the square root X, and it is from the range of 0 to 1. Okay, so here's what we do. We just take the integral from that region and calculate for all of them and then see if these areas come out to be the same. And if you integrate this, you'll have to substitute or square of X as you and then your do you would be equal to one over to swirl x dx We know when x zero u zero X when you use one and then we have the X Thanks is equal to two root X Do you? We know dx also be due to you do you right? And then if we go on, substitute those values in. We have the integral films. You're the one you to you. Do you just equal choose. She would want to You you Do you okay now? This is actually interesting. If you go and look at your second graf, you can see that it has the function written on there. It has a function Why is equal to to x x which really is the same thing as to you you because we've just substituted those values said. And if we were to take the integral Excuse me, I mean to say if we were to take the integral 01 we would have the same thing as to you. You you which are both equal to each other. Therefore, we can confirm that one and two are equal to each other now. The third integral runs from zero up. I have it ISS the integral e to sign X sign two X is equal to zero Hi over to need to the sine x to sign ax cozy tax then should simplify this It will come out to be integral zero to pi How sign to sign like speed sine x I was Kassian x t x And if we let sine x equal to you here do you would be with your clothes on X dx and if you went ahead and we know that when x zero. You is your own excess pirate to use one. So we went ahead and substituted these values we would get. So you wrote a one to you. You do you? Which is the same thing as this one, This one. So we can confirm that one is equal to two. And also equal to the third graf. All right. So we can confirm that the areas of all three graphs are the same. All right, well, I hope that clarifies the question. Basically, all we did is take the integral from the regions that were given for each graph. And then we compared them all three. And we noticed that there were all the same. That's how we got our final answer. That all three. All right, well, thank you for watching.

The University of Texas at Arlington