Suppose the total area under the curve y=x^2-mx+1 from 0 to 5 is 11.3024. Find m.
I know that this means that i need to find the integral of the absolute value of x^2-mx+1 from 0 to 5 and set it equal to 11.3024, but I am not sure where to go from there.
so area
= integral (x^2 - mx + 1) dx from 0 to 5
= x^3 /3 - (m/2)x^2 + x from 0 to 5
= (125/3 - 25m/2 + 5) - (0-0+0)
but this equals 11.3024
times 6
250 - 75m + 30 = 67.8144
-75m = -212.1856
m = 2.8291
I am trying to find the total area under the curve, not just the integral...
You know the roots are
x = (m±√(m^2-4))/2
So, if m<=2 there are not two real roots, so the total area is the same as the integral.
If m>2, then you need to divide the integral into 3 parts, with the roots as the limits of integration.
Reiny's solution of m=2.8291 indicates to me that you will need to break up the integral, since the curve dips below the x-axis.
To find the value of m, you need to evaluate the definite integral of the function y = x^2 - mx + 1 from 0 to 5 and set it equal to 11.3024. Here's how you can do that:
1. Start by finding the indefinite integral of the function y = x^2 - mx + 1:
∫ (x^2 - mx + 1) dx = (1/3) x^3 - (m/2) x^2 + x + C
2. Next, use the definite integral property to evaluate the definite integral from 0 to 5:
∫[0 to 5] (x^2 - mx + 1) dx = [(1/3) x^3 - (m/2) x^2 + x] [0 to 5]
3. Substitute the upper limit (5) into the integral expression:
[(1/3) (5)^3 - (m/2) (5)^2 + 5] - [(1/3) (0)^3 - (m/2) (0)^2 + 0]
4. Simplify the expression:
[(125/3) - (25/2)m + 5] - [0]
5. Combine like terms:
(125/3) - (25/2)m + 5 = 11.3024
6. Rearrange the equation and solve for m:
(125/3) - (25/2)m = 11.3024 - 5
7. Simplify the right side:
(125/3) - (25/2)m = 6.3024
8. Get a common denominator:
(250/6) - (75/6)m = 6.3024
9. Combine the fractions:
(250 - 75m) / 6 = 6.3024
10. Multiply both sides of the equation by 6 to eliminate the fraction:
250 - 75m = 6 * 6.3024
11. Calculate the right side:
250 - 75m = 37.8144
12. Move the constant term to the other side:
-75m = 37.8144 - 250
13. Simplify the right side:
-75m = -212.1856
14. Divide both sides by -75 to solve for m:
m = (-212.1856) / -75
15. Calculate the value of m:
m ≈ 2.8291
Therefore, m is approximately 2.8291.