Use a line integral to find the area of the region R. R: region bounded by the graphs of y=2 x+1

Use a line integral to find the area of the region R. R:...

Key Concepts

Intersection of Curves

Finding the intersection points of curves is essential in determining the boundaries of a region. These points establish the limits of integration and are key to ensuring that the entire area of the region, as defined by the intersecting curves, is accurately captured by the integral.

Green's Theorem

Green's Theorem provides a connection between a line integral around a simple closed curve and a double integral over the region bounded by that curve. It is especially useful for converting a difficult double integral into a more tractable line integral, which can simplify the calculation of quantities like area by selecting appropriate functions.

Area Calculation via Line Integrals

Area calculation via line integrals involves setting up an integral over the boundary of the region such that the integral evaluates to the area inside the closed curve. This is achieved by choosing specific vector fields, where the integral using Green's Theorem simplifies the evaluation of the enclosed area.

Parametrization of Curves

Parametrization of curves is the process of representing a curve with functions of a parameter, typically making integration along the curve manageable. This step is crucial when applying line integrals as it converts the integral into a definite integral with respect to the parameter over the interval that represents the curve.

Transcript

00:01 In this question, we need to calculate the area of the region bounded by the curves using the line integral.

00:09 So the curves are given as y equals to 2x plus 1 and y equals to 4 minus x square.

00:15 So first of all, we will sketch the graph of these equations.

00:20 So here we sketched and sketched the graphs of these two equations and also shaded the region.

00:26 We need to find the area of.

00:28 So, we will use the line integral to calculate this area given by these curves.

00:36 So we know that area can be calculated as half times the integral, there is line integral over the curve c of x .d .y minus y.

00:48 D .x.

00:54 Now, we know we can use a green's theorem to calculate such an integral that is against theorem states that the integral of over the that is line integral over the curve c of m.

01:11 D x minus n.

01:15 D .y can be calculated as the double integral over the region r of del n over del x minus del m over del y.

01:28 .de a where a is the area bounded by that reason.

01:36 Now, so for the given expression that is the given integral, we can write the values of m and n, that is, so m will be minus y and n will be x from here.

01:57 M will be minus y and n will be x.

02:01 So from here we can calculate that area as area will be equal to half times.

02:08 This integral will be replaced by the double integral or double integral over the region r of del n over del x del n over del x will be that is del x over del x which will be one minus del m over del y we know that m here is minus y so del n over del y will be minus one so one minus minus one dot d a now we know that da can be replaced by dx, dy...

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