Solve the given problems by integration. Find the first-quadrant area bounded by y=(ln (4

Solve the given problems by integration. Find the...

Key Concepts

Area Under a Curve

The concept of the area under a curve is central in calculus, where integration is used to calculate the space between a function and the x-axis over a given interval. It is particularly useful in problems that ask for the determination of regions bounded by curves, where the integral represents this area.

Integration by Substitution

Integration by substitution is a technique used to simplify an integral by introducing a new variable that transforms the original integral into a simpler form. This method is particularly effective when the integrand contains a composite function or when a transformation of variables results in a more manageable integration process.

Logarithmic Functions in Integration

Logarithmic functions often appear in integrals and require careful manipulation and substitution to simplify the integrals. When dealing with expressions involving logarithms, one must often use properties of logarithms in combination with integration techniques such as substitution to handle the more complex integrands effectively.

Limits of Integration

Limits of integration define the endpoints of the interval over which the integration is performed. They are crucial in definite integration, as they ensure that the area is calculated only over the specified region, corresponding to the boundaries of the problem, such as restrictions to certain quadrants.

Definite Integration

Definite integration involves calculating the integral of a function over a specific interval, which represents the accumulation of quantities such as area. In problems involving areas, the definite integral sums up infinitely many infinitesimal contributions, thus computing the total area under the curve over a bounded region.

Transcript

00:01 In this question, we want to find the area under this curve bounded in the first quadrant and x equals to 5.

00:09 Now in general, for a general graph, y equals to fx, when we want to find area under the curve from x equals to a to b, we will take a small strip over here.

00:29 The width is so small, it will be delta x or dx.

00:34 And the height over here is y.

00:36 So the area for this strip is y -d -x.

00:40 Now to find area under the curve from this portion to this portion over here will be summing up all these strips.

00:48 So it will be a continuous summation from x equals to a to b.

00:52 So you can see that this is the area under curve.

00:58 So in this case, the area under this curve in the first quadrant will be from x equals to 0 to 5, y -d -x.

01:06 So the y is this, the x.

01:14 There are a couple of ways to integrate this, but for this topic, we are using substitution method and the integration power rule.

01:22 So let you be lawn 4x plus 1.

01:28 So the u over the x will be differentiate the lawn.

01:33 I'll have 1 over 4x plus 1, and then differentiate inside the bracket.

01:38 I'll get 4.

01:39 When i differentiate this, 4x, i'll get 4 plus.

01:43 When i differentiate 1, it'll just be 0 as it is a constant.

01:47 So, du will be 4 over 4x plus 1, dx.

01:53 Now you can see that we have the dx here.

01:56 We have the 4x plus 1 over here.

01:57 So we are missing a 4 over here in this expression.

02:02 So let's adjust it a little bit.

02:05 So to get a 4, let me put a quarter here and i'll put a 4 over here.

02:10 So my over 4x plus 1 and the x will be at the back here and my lawn 4x plus 1 will be over here.

02:23 Now you can see that this portion here is my u and this portion here is my du.

02:32 Now don't forget to sub up the x equals to 0 and x equals to 5 with the u.

02:36 So when x equals to the lower limit on 0, you will be long 0 plus 1 and that will be 0...

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