Sketch the region enclosed by the given curves and find its area. y = (x)/(1 + x^2) , y = (x^2)/

Sketch the region enclosed by the given curves and find its...

Key Concepts

Intersection Points

Determining where the curves intersect is a crucial step because these points establish the limits of integration when calculating the area between curves. This involves setting the equations equal to each other and solving for the variable, which provides the boundaries of the region of interest.

Area Between Curves

The area of a region bounded by two curves is found by integrating the difference between the upper (or outer) function and the lower (or inner) function over the interval defined by their intersection points. This concept relies on setting up a proper definite integral where the integrand represents the vertical distance between the curves.

Definite Integration

Definite integration is a fundamental tool in calculus used to calculate the accumulation of quantities, such as areas under or between curves. It applies the interval limits (often determined by intersection points) to compute the exact area using the integral of a function over a specified range.

Graphical Analysis

Sketching the curves to visualize the region enclosed helps in understanding the relative positions of the functions, which is essential to correctly identify which function acts as the upper or lower boundary on the interval of integration. This facilitates the proper setup of the integral for calculating area.

Transcript

00:01 We want to do a quick sketch of these two equations, and then we want to determine the area of the enclosed region.

00:08 So let's take a look at our first one.

00:13 So we know that as x goes to infinity, and also x goes to negative infinity, we know that the asymptotic behavior is that it goes to zero.

00:22 So for this one, we know that we're going to zero, both on the right and the left.

00:31 And we also know that it's going to cross the origin.

00:34 What else do we know? we know that the denominator, 1 plus x squared, is always greater than the numerator for all x.

00:42 So we're never going to pass the values between minus 1 and 1.

00:49 So this is what the equation looks like.

00:54 Now let's take a look at the second one.

01:00 For the second one, we also see the same behavior as x goes to plus and minus infinity.

01:05 We see that it's going to 0 because x cubed is in the denominator.

01:09 It's greater than x squared.

01:11 But we also have this one vertical asymptote, since the denominator can be 0, at x equals negative 1, since negative 1 cubed is minus 1.

01:21 So we should draw in that asymptote.

01:26 This is at x equals minus 1.

01:30 Let's draw this one in blue.

01:33 So to the left, we see that x is going to minus infinity and it goes to 0.

01:40 But on the right, we also see that this function must touch the origin and it exhibits this kind of behavior.

01:48 It kind of goes up and then it goes back down.

01:52 So we do see that there's this enclosed region right here.

01:57 And that's the area that we want to determine.

02:00 So we need to figure out our limits of integration.

02:03 We know one will be the origin and we need to figure out the other.

02:06 So to solve for these, we equate the two equations.

02:09 So let's do that.

02:13 So our first one was x over 1.

02:16 Plus x squared.

02:19 The second one was x squared over 1 plus x cubed.

02:26 So we're just going to multiply out the denominators.

02:30 So here we have x times 1 plus x cubed equals x squared times 1 plus x squared.

02:40 Okay, x plus x to the 4 equals x squared plus x to the 4.

02:53 Okay.

02:55 So we see that these x the 4 is cancel.

02:59 We bring everything to one side.

03:01 X squared minus x equals x times x minus 1.

03:05 So we see that our values are x equals 0, which we expected, and x equals 1.

03:13 So we see that this point over here is x equals 1, and this one is x equals 0.

03:19 So we're just going to integrate from x equals 0 to x equals 1.

03:25 Area equals integral 0 to 1...

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