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

Key Concepts

Intersection of Curves

This concept involves finding points where two functions meet by equating their expressions and solving for the variable. Determining these intersection points is essential because they define the limits of the region over which the area is computed.

Sketching Functions

Sketching functions aids in visualizing the curves, understanding their relative positions, and identifying the bounded region. Recognizing the shape and key features of functions such as square root curves and linear equations is fundamental for setting up the problem.

Definite Integration

Definite integration is the process used to calculate the area under a curve or between curves over a specific interval. By integrating the difference between the upper and lower functions within the intersection limits, one obtains the area of the enclosed region.

Setting Up Integration Bounds

Once the intersection points are determined, these values serve as the boundaries (limits) for the integral. Correctly establishing these limits is crucial for accurately computing the area of the region defined by the curves.

Transcript

00:03 In this question, we're asked to find the area between two curves.

00:09 The area looks like this in the blue region.

00:13 Ignore the graph underneath it.

00:16 So the first step is we need to find the intersection points.

00:28 We know that they're going to intersect at zero.

00:34 This is simply by observation.

00:38 We can alternatively set the two root quantities equal to each other, and then try and solve.

00:47 And then try and solve that equation through there.

00:58 In other words, we're going to have, also by observation, you'll notice that x equals 1 will also fit in.

01:18 So both of these can be really found by observation, and you don't really need to solve any equations.

01:26 So it's also obvious that the area that the top curve is x times x squared, the square of x squared plus 1.

01:35 And the bottom curve is the other curve.

01:38 And we know that over some region, we know that the area between them is given by integrating the top curve minus the bottom curve.

01:50 So let's plug in all of this information now.

01:53 That means that we have the integral from 0 to 1.

01:59 We'll have x times the square root of x squared plus 1.

02:04 And we're going to actually split this into two integrals, just so we make our life easier.

02:09 Because both of these integrals will have to use a u substitution.

02:16 Both of these i will do in green and blue respectively.

02:22 So for the one in green, i'm going to write it down here.

02:34 We're going to let you be equal to x squared plus 1.

02:38 Du will be equal to 2x dx, and by solving for dx, we'll get that dx is du over 2x...

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