Sketch the region enclosed by the given curves and find its area.
$ y = x ^3 $ , $ y = x $
Applications of Integration
All right. Now sketch the region enclosed by those two curves. So this is our XY plane. And the first function is why he goes to X Cube. So you should look like this. This is our by equals two x Q And why he goes to X. Okay, it's a street line. This is why it goes x in this those two eco region since everything is symmetric, uh, with respect or region. So those two are exactly the same. Um, they have the same area, and now we need to find this shaded region area. Um, so they will be, um yeah, we only need to calculate one side. So wait time and then times two. So it will be two times. Um, this shaded region. Yeah. Um, and, uh, I want to calculate this region by taking integral with respect to X. Since everything is represented bags All right. Right. The next step, we need to find the boundary. Correct. Clearly, X goes from zero to this point this point x two all right and x two should satisfy the those two functions, which is X square equals two x, so x two should satisfy this equation. Yeah. Since extra is not zero, we can cancel one X Then we got X squared equals one, which gives us X two eco swat. Since X two is part, so are integral. That goes from zero to one. Okay, um, so then the thing inside the integral is the upper curve, minus the lower curve. So our upper curve here is a straight line X minus glower, curve xq. Yeah. Then let's find anti duality of that. That will be up half black Square, minus fourth. Thanks for you had a boundary one and zero. So Well, X equals one we plug in. This will give us one half minus or quarter, which is a quarter times two is about half right minus, we know, actually called zero. This term, zero times two is also zero. So it's like one half minus zero. Our answer will be just 7.5