Suppose f(x)=2x^2-x^3 and g(x)= sin(pix/2).
a) What is the exact value of the integration from 0 to 2[f(x)-g(x). Find numerical approximation of this value. What does the value of this integral tell you about the areas of the regions beteen the two graph
I am uncertain of your questions about the three things you are asked to do.
The first is easy
The second you have to decide on the numerical model, I would use .1 divisions of x.
On the third, compare the value of the integral of 2(f(x)-g(x)) to the value of INT sin(PIx/2). That will be revealing.
Repost if you have a specific question.
To find the exact value of the integration from 0 to 2 of [f(x) - g(x)], we need to first find the antiderivative (or the integral) of each function.
For f(x) = 2x^2 - x^3, we can find its antiderivative using the power rule for integration. The antiderivative of 2x^2 is (2/3)x^3 and the antiderivative of -x^3 is -(1/4)x^4. Therefore, the antiderivative of f(x) is (2/3)x^3 - (1/4)x^4.
For g(x) = sin(pix/2), the antiderivative involves using trigonometric identities. The integral of sin(ax) with respect to x is -cos(ax) / a. Therefore, the antiderivative of g(x) is -2cos((π/2)x) / π.
Now, let's calculate the definite integral from 0 to 2 of [f(x) - g(x)]. Substituting the limits, we have:
∫[f(x) - g(x)] dx = [(2/3)x^3 - (1/4)x^4 - (-2cos((π/2)x) / π)] evaluated from 0 to 2.
Evaluating the expression at x = 2, we get:
[(2/3)(2)^3 - (1/4)(2)^4 - (-2cos((π/2)(2)) / π)] = 16/3 - 8 - (-2cos(π)) / π = 16/3 - 8 + 2 / π
Now, to find the numerical approximation of this value, we can use a graphing calculator or software that provides integral calculation capabilities. By inputting the function [f(x) - g(x)] and the limits 0 and 2, we can obtain the numerical approximation.
The value of this integral represents the signed area between the graphs of f(x) and g(x) from x = 0 to x = 2. The fact that it is an integral of the difference of the two functions means that it calculates the net area, taking into account the regions where f(x) is either above or below g(x). If the integral is positive, it indicates that the area between the functions is positive, meaning f(x) is mostly above g(x) in that interval. If the integral is negative, it indicates that the area between the functions is negative, meaning f(x) is mostly below g(x) in that interval.