f(x)=x^2 - 5x +7 and g(x) 3/x
1) Indicate points of intersection
2) compute the area enclosed by the graphs of f(x) and g(x).
1) (1, 3) , (3, 1)
2)Area = ∫ [1 to 3] [(3/x) - (x^2 - 5x + 7)] dx
= [3 ln x - (1/3)x^3 + (5/2)x^2 - 7x
= 3 ln 3 - 8/3
am i doing the right thing here?
Looks to me like you've got it.
Yes, you have done the right computations to find the points of intersection and compute the area enclosed by the graphs of f(x) and g(x).
For the first question, to find the points of intersection, you need to solve the equation f(x) = g(x). In this case, you have the equations f(x) = x^2 - 5x + 7 and g(x) = 3/x. Setting these equations equal to each other, you get:
x^2 - 5x + 7 = 3/x
Multiplying both sides by x to clear the fraction, you get:
x^3 - 5x^2 + 7x - 3 = 0
This is a cubic equation, and you can solve it using various methods such as factoring, synthetic division, or numerical methods like Newton's method. By solving this equation, you found that the points of intersection are (1, 3) and (3, 1).
For the second question, to compute the area enclosed by the graphs of f(x) and g(x), you need to find the definite integral of the absolute difference between the functions f(x) and g(x) over the interval from x = 1 to x = 3. This can be represented as the following integral:
Area = ∫[1 to 3] [(3/x) - (x^2 - 5x + 7)] dx
Evaluating this integral, you have:
Area = [3 ln x - (1/3)x^3 + (5/2)x^2 - 7x] from 1 to 3
Substituting the limits of integration:
Area = [3 ln 3 - (1/3)(3^3) + (5/2)(3^2) - 7(3)] - [3 ln 1 - (1/3)(1^3) + (5/2)(1^2) - 7(1)]
Simplifying this expression, you will get the value:
Area = 3 ln 3 - 8/3
So, yes, your computation is correct. The area enclosed by the graphs of f(x) and g(x) is 3 ln 3 - 8/3.