Question 693472: the rear stabilizer of a certain aircraft can be described as the region under the curve y=3x^2-x^3 as shown in figure. What is the area of the rear stabilizer?
Answer by tinbar(133)   (Show Source):
You can put this solution on YOUR website! Since I cannot see the figure as implied by the question, I cannot guarantee the correctness of the following answer, though I expect it is the appropriate solution
You need to perform an integration. First you find that the curve cuts the x axis at x=0 and x=3
The x-axis as a function of y is just y = 0. So now you need Integral((f(x) - 0)dx) = Integral(f(x)dx) where the limits are x = 0 to x = 3, and here f(x) is the function you gave.
The anti-derivative of f(x) is x^3 - (1/4)*x^4, which we will call F(x). To see this is true we check that F'(x) = f(x). I will leave it to you to confirm this.
Now we compute F(3) - F(0) to find the desired area, which turns out to be F(3) = 3^3 - (1/4)*(3^4) = 27 - (1/4)*(81) = 27/4.
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