Integration Applications: find the area enclosed by the following curves:- Y= lnx and y= -2x+3 and the ordinate x=3

Integrate lnx +2x - 3 dx from x = a to x = 3.

a is the value of x where the two lines intersect. You will have to solve for that numerically. It is about 1.34

To find the area enclosed by the curves y = lnx and y = -2x + 3, as well as the ordinate x = 3, we need to set up an integral that represents the area between these curves.

First, we need to determine the x-coordinate where the two curves intersect. To find this point, we set the equations for both curves equal to each other:

lnx = -2x + 3.

Rearranging this equation, we get:

lnx + 2x = 3.

Now, let's solve this equation to find the x-coordinate where the curves intersect.

Unfortunately, there is no algebraic way to solve this equation, so we will need to use an iterative numerical method, such as the Newton-Raphson method or trial and error, to find an approximate solution. Let's assume that the x-coordinate of the intersection point is x = c.

Now, we will set up the integral to find the area enclosed by the curves. Since the curves intersect at x = c, the limits of integration will be from x = 0 to x = c. The area between the curves is given by:

Area = ∫[0, c] (lnx - (-2x + 3)) dx.

Simplifying this integral, we get:

Area = ∫[0, c] (lnx + 2x - 3) dx.

Next, we integrate the expression inside the integral:

Area = [xlnx + x^2 - 3x] evaluated from 0 to c.

Substituting the limits, we have:

Area = (clnc + c^2 - 3c) - (0ln(0) + 0^2 - 3(0)).

Since ln(0) is undefined, the term 0ln(0) is equal to 0. Hence, the second part of the equation disappears, leaving us with:

Area = clnc + c^2 - 3c.

To find the specific value of the area, we need to determine the x-coordinate where the curves intersect (x = c).

Now, let's find the value of c by solving the equation ln(c) + 2c = 3, using numerical methods like the Newton-Raphson method or trial and error.

Once you have found the value of c, substitute it back into the area equation (clnc + c^2 - 3c) to get the final value of the enclosed area.

Please note that finding the exact value of the area might require the use of numerical methods, as mentioned before.