Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x an

step 1 So for this question, let's draw those curves first. So here is xxy axis. And the first function

Sketch the region enclosed by the given curves. Decide...

Key Concepts

Intersection of Curves

Determining the intersection points involves solving the equations of the curves simultaneously. These intersection points define the limits of integration and are essential for accurately describing the region whose area is to be calculated.

Area Between Curves

The area between curves is evaluated by integrating the difference between the functions that bound the region. This concept requires identifying which function lies above (or to the right of) the other over the interval of integration, ensuring that the area is computed as the positive difference between the curves.

Order of Integration

Choosing whether to integrate with respect to x or y depends on the geometry of the region and the ease of expressing the boundary curves in one variable compared to the other. This decision simplifies the setup of the definite integral by providing clear, manageable limits.

Approximating Rectangles (Riemann Sums)

Dividing the region into small, typical rectangles is the foundational idea behind Riemann sums. As the width of these rectangles approaches zero, the sum of their areas approaches the exact area of the region, which is formally defined by the definite integral.

Transcript

00:01 So for this question, let's draw those curves first.

00:07 So here is xxy axis.

00:12 And the first function, y equals x minus 2 squared.

00:18 So it's actually y equal to x squared shift to the right by 2.

00:23 So here is 2, and the function will just look like x squared.

00:34 Okay, and y equal to x is a line, a straight line.

00:43 So this is our y equals to x, and the first line is y equal to x square.

00:54 Okay, so our area are enclosed regions here.

01:05 Alright, so since everything is represented by x, we will integrate with respect to x.

01:12 And to draw a typical approximating rectangle, so say we draw a rectangle right here, and we zoom out this rectangle, so this will be our typical approximating rectangle, it's a rectangle, right? so the wealth will be delta x, and the height, the height will be the upper curve minus the lower curve.

01:44 So the upper curve is xi, and our lower curve here is x i square so x i minus x i square then we can write down our formula for the area of this region area of this region a will equals to integral with respect to x and we need to find the boundary for x we can see from the picture the boundary start from here x1 here x2 right so we know our integral goes from x1 to x2 and we'll figure out those x1 and x2s later and the integral we put in is just this height here it's x i minus x i square so when we put here it's just x minus x squared now let's try to find x1 and x2 so we know x1 x2 are just the intersection for those two curves so we put them to so which means our x i should satisfy both equations in other words x i if we put into the oh sorry i made a mistake here this is x minus 2 to the square so actually here is so here is x minus 2 squared right so the height will be x x i minus x i minus 2 squared and here we change it to x minus 2 squared right and to find those x1 and x2 we plug into the equation so we basically we got x i minus 2 squared equals to x i in other words the typical approximating rectangle at x1 and x2 the height will be 0 right as you can see that the shrink down to no height here is just a dot in fact it's just a dot so there's no height in other words, this equal to 0.

04:55 So this will give us this formula right there.

04:58 Then we can solve for x1 and x2.

05:01 X1 is the smaller one and x2 is a bigger one.

05:05 So let's do that...

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