Area of a region between curves Find the area of the entire region bounded by the curves y=(x^3)

step 1 suppose we have the two curves. Why? Equal to X Cubed over X squared plus one hand. Why equal Te

Area of a region between curves Find the area of the entire...

Key Concepts

Intersection of Curves

Determining the points where two curves intersect is essential because these points define the limits of integration when calculating the area of the region enclosed by the curves. This typically involves setting the functions equal to each other and solving for the variable.

Definite Integration

Definite integration is used to compute the area under a curve over a specific interval. When finding the area between curves, the integral of the difference between the upper curve and the lower curve over the interval defined by their intersection points is evaluated.

Area between Curves

The concept involves calculating the total area of the region bounded by two curves by integrating the difference (upper curve minus lower curve) across the range defined by their intersection points. This requires both establishing which function is on top over the interval and performing the integration accordingly.

Algebraic Simplification

Before integrating, it is often necessary to simplify the expression for the difference between the functions. This step helps in making the integration process manageable and may involve factoring, combining like terms, or cancelling common factors.

Transcript

00:01 Suppose we have the two curves.

00:04 Why? equal to x cubed over x squared plus one hand.

00:08 Why equal teoh eight acts over expert most wanted and we want to find the area of the region bounded by these two curves.

00:18 So you might be daunted and trying to graft these because of this x squared plus one on the denominator.

00:23 But they both have the same denominator and notice that x squared plus one is always positive for any value of x because x squared is never negative and ones always positive.

00:40 So from here we can just look at the graphs of x cubed and eight x to kind of get an idea of what that region is going to look like.

00:50 So i graph x cubed.

00:53 It's kind of like so forgive my drawing.

00:57 And if i graph a x, it's gonna be a steep line.

01:06 So the region were looking for is gonna be aware these two curves meat and since he's the left and right hand side of the uae, access are going to be symmetric or really ah, mirror.

01:31 What we can do is we can just find the area of this region then multiply by two.

01:41 So where did they meet? so we want to know that.

01:43 So if we said execute equal to eight acts that'll tell us where they meet obviously mean it.

01:49 Zero this week.

01:50 In fact, your ex out.

01:51 So we have x.

01:54 Actually, let's just let's pull both on one side and solve.

01:58 So x cubed minus eight x equals zero.

02:05 So we have x times x squared minus eight people zero.

02:14 So our solutions will be x equals zero through eight negative route eight or you can write these is too rude to either way is fine.

02:24 So now we just need to set up the inner role and we can tell by the graph that, um, eight x is greater than execute.

02:32 So we have a negative x cubed plus x over x squared plus one from zero to root eight dx.

02:47 Now what do we dio? so we need teoh reduced this polynomial in some way and a good way to do that is going to be doing ah put take so we have room hates if x cubed.

03:05 I want i want to break this into the execute one and the eight x so have.

03:16 So if i want this to be divisible by x squared plus two that i need a minus x here.

03:28 Sorry, export plus one.

03:30 So then this would be equal to negative acts.

03:33 So if i subtracted x here, i need to add it back later.

03:38 We can add it back into the other ex plus x plus eight x over x squared plus one dx...

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