Calculate the area bounded by the x-axis and the function f(x)= -(x-a)(x-b), where a<b and a and b are constants. Please do it out in steps so I can understand it, and try to simplify the final answer and leave it in factored form.
Frankly, I would multiply out the polynomial, then integrate
f(x)= -(x-a)(x-b)
= -x^2 + ax + bx -ab
Now integrate it.
I will be happy to critique your work.
To calculate the area bounded by the x-axis and the function f(x) = -(x-a)(x-b), we need to find the integral of the function over the appropriate limits. Let's go step by step:
1. First, expand the function and simplify:
f(x) = -(x-a)(x-b)
= -x^2 + ax + bx - ab
2. Next, integrate the function with respect to x:
∫ f(x) dx = ∫ (-x^2 + ax + bx - ab) dx
To integrate, we apply the power rule for each term:
∫ -x^2 dx = - (1/3) x^3 + C1 (integrating -x^2 gives -(1/3) x^3)
∫ ax dx = (a/2) x^2 + C2 (integrating ax gives (a/2) x^2)
∫ bx dx = (b/2) x^2 + C3 (integrating bx gives (b/2) x^2)
∫ -ab dx = -abx + C4 (integrating -ab gives -abx)
3. Combine the integral results:
∫ f(x) dx = - (1/3) x^3 + (a/2) x^2 + (b/2) x^2 - abx + C
4. Now, we need to evaluate the integral between the appropriate limits. Since we are calculating the area bounded by the x-axis, the limits of integration will be the x-values at which the function intersects the x-axis.
The function intersects the x-axis when f(x) = 0, which occurs when x = a and x = b. So, we integrate the function within these limits:
Area = ∫[a,b] f(x) dx
= ∫[a,b] (-x^2 + ax + bx - ab) dx
= ∫[a,b] -x^2 dx + ∫[a,b] ax dx + ∫[a,b] bx dx - ∫[a,b] ab dx
5. Evaluate the definite integrals:
∫[a,b] -x^2 dx = - (1/3) [b^3 - a^3]
∫[a,b] ax dx = (a/2) [b^2 - a^2]
∫[a,b] bx dx = (b/2) [b^2 - a^2]
∫[a,b] ab dx = ab(b - a)
6. Substitute the evaluated definite integrals into the area equation:
Area = - (1/3) [b^3 - a^3] + (a/2) [b^2 - a^2] + (b/2) [b^2 - a^2] - ab(b - a)
= - (1/3) b^3 + (1/3) a^3 + (1/2) a b^2 - (1/2) a^2 b - (1/2) a b^2 + (1/2) a^2 b + (1/2) b^3 - (1/2) a b^2
= - (1/3) b^3 + (1/3) a^3 - (1/2) a b^2 + (1/2) a^2 b + (1/2) b^3 - (1/2) a b^2
= - (2/6) b^3 + (2/6) a^3 - (3/6) a b^2 + (3/6) a^2 b + (3/6) b^3 - (3/6) a b^2
= (1/6) a^3 - (1/6) b^3 + (1/3) a^2 b - (1/3) a b^2
Finally, the simplified and factored form of the area bounded by the x-axis and the function f(x) = -(x-a)(x-b) is (1/6) (a^3 - b^3 + 2a^2 b - 2a b^2).