20. Is the function f (x.v) = 2x(r? +y2 10)-' integrable over [0. 1] [0. 1J? Over [-3.3] / [-3.3J? Over [~I0. 10] x [~10, 10J?...

Assume $f$ and $g$ are even, integrable functions on $[-a, a],$ where $a>1 .$ Suppose $f(x)>g(x)>0$ on $[-a, a]$ and that the area bounded by the graphs of $f$ and $g$ on $[-a, a]$ is $10 .$ What is the value of $\int_{0}^{\sqrt{a}} x\left[f\left(x^{2}\right)-g\left(x^{2}\right)\right] d x ?$

Hello, everyone. We're going to find the numerical integral off function. F F X is equal to over one plus expert Indian in the interval, 0 to 10 to the eighth power. Um and so we're going to be using the end I auntie function on the calculator, which stands for the an Integral. And we're gonna plug in this information along the function, as was Interval and they'll tell us the area under the curve, which is just integral. So when we do so and we look, enter, we get their answer is going to be equal to 3.1416 Um And so, interestingly enough, this is quite close to pipe by 3.14159 But there's something quite interesting. Anyways, the integral is going to be for the area under the curve of dysfunction is going to be 3.1416 And we found this using that end in function on a calculator. So thank you for watching. And I hope this helped

Became for this problem. We're just asked to find a miracle and groom of f of X equals X squared on interval. Thank you for this year. All right, this one simple. We just used the Miracle Gro on a calculator and we plug in that squared X, which tells us well what it tells the calculator what variable we're using. And then from negative 4 to 0, they're simple. We get an answer of 21.33 your opinion.

Hello, everyone. We're using the end end on a calculator to find a new miracle integral off the function over the specified interval. So we have FX is equal to X squared and the interval is from 04 So when we put this into a cab later, the calculator will take the integral in this interval from 04 which we just specify. So upon doing so we get that answer secret or 21.33 And if you want to convert that into a fraction, you go to Matt, then frak. And it will convert that to a fraction for you, which in a couple of buttons you see that it's 64 over three. And so thank you for watching, and I hope so.

Okay. This question tells us some integral values along with f of X is behavior on these intervals. And it's gonna have us use this information to evaluate some similar looking into girls. So part A once the integral from 0 to 5 of after backs and we don't have an outright expression for this. But we do know that it's the same thing as first collecting area from 0 to 2, then adding that with the area we collect from 2 to 5 and these values air known, this integral is six. And this in a Kroll is negative eight. So we have negative too. Then for part B, we want the integral from 0 to 5 of absolute value, half of x dx. And from here is where we look at the function behavior because we know the F of X is always greater than equal to zero in the first interval and always less center equal to zero in the second. So the first the first part doesn't change. But with the second part, all that negative area becomes positive. So all we dio is subtract that integral to flip the sign and again we said this Integral was six and this integral was negative. Eight. So we have 14 total and all that absolute value did was make all that negative area in the second part into a positive area, then part C. Once the inter grow from 2 to 5 of four times the absolute value of F of X dx and we can pull out that four in front of our integral because it's just a constant. And then we already said with the absolute value of f of X did on that interval, it just changes all the negative area we have to a positive. And since we know that all the area is initially negative, it just flips the sign of that. So it's four times eight, which is 32 then. Lastly, it wants the integral from 05 of f of X plus absolute value of f of x, d x, and we'll split this up into two into grows with the addition sign. So first we have the normal and then we have the absolute value and we already actually found both of thes. So he said this first integral was equal to negative two in the second, integral was equal to 14 so the result is 12