Asked by Wawen
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Find the area below f(x) =-x^2 + 4x +3 and above g(x) = -x^3 +7x^2-10x + 5 over the interval of [1 ≤ x ≤ 2]. Show the overlap or shaded area of the graph.
Find the area below f(x) =-x^2 + 4x +3 and above g(x) = -x^3 +7x^2-10x + 5 over the interval of [1 ≤ x ≤ 2]. Show the overlap or shaded area of the graph.
Answers
Answered by
Reiny
let's take a look what we are dealing with
http://www.wolframalpha.com/input/?i=plot+y+%3D-x%5E2+%2B+4x+%2B3+,+y+%3D+-x%5E3+%2B7x%5E2-10x+%2B+5
We can see that within your domain of 1 ≤x≤2 , they do not cross each other.
so the effective height of our region
= -x^2 + 4x +3 -( -x^3 +7x^2-10x + 5)
= -x^2 + 4x +3 + x^3 -7x^2 + 10x - 5
= x^3 - 8x^2 + 14x - 2
area = ∫(x^3 - 8x^2 + 14x - 2) dx from 1 to 2
= [(1/4)x^4 - (8/3)x^3 + 7x^2 - 2x] from 1 to 2
= ....
you do the arithmetic, you should get 49/12
http://www.wolframalpha.com/input/?i=plot+y+%3D-x%5E2+%2B+4x+%2B3+,+y+%3D+-x%5E3+%2B7x%5E2-10x+%2B+5
We can see that within your domain of 1 ≤x≤2 , they do not cross each other.
so the effective height of our region
= -x^2 + 4x +3 -( -x^3 +7x^2-10x + 5)
= -x^2 + 4x +3 + x^3 -7x^2 + 10x - 5
= x^3 - 8x^2 + 14x - 2
area = ∫(x^3 - 8x^2 + 14x - 2) dx from 1 to 2
= [(1/4)x^4 - (8/3)x^3 + 7x^2 - 2x] from 1 to 2
= ....
you do the arithmetic, you should get 49/12
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