Find the area enclosed between the ffg curves :- y=lnx and y= -2x+3 and the ordinate x=3
To find the area enclosed between the curves, we will integrate the difference between the two curves with respect to x.
Step 1: Find the points of intersection:
Set the two equations equal to each other and solve for x:
lnx = -2x + 3
We can solve this equation numerically or graphically. A graphical solution can be obtained by plotting both curves on the same graph and finding their intersection. It turns out that there are two points of intersection: x ≈ 1.276 and x ≈ 2.732.
Step 2: Set up the integral:
To find the area, we need to integrate the difference between the two curves from the leftmost intersection point to the rightmost intersection point:
∫ [f(x) - g(x)] dx
Let's denote f(x) as the upper curve (-2x + 3) and g(x) as the lower curve (lnx). Therefore, our integral becomes:
∫ [(-2x + 3) - (lnx)] dx
Step 3: Evaluate the integral:
Integrate the expression with respect to x from the first intersection point to the second:
∫[(-2x + 3) - (lnx)] dx (limits: x ≈ 1.276 to x ≈ 2.732)
This integral can be evaluated using calculus or numerical methods. The result will give us the area enclosed between the curves.
Step 4: Substitute the upper limit (x = 3):
We also need to subtract the area under the curve for x > 3 (to the right of the vertical line x = 3). To do this, we substitute x = 3 into the upper curve equation (-2x + 3) and subtract it from the previous result.
This will give us the final area.