This is just like the last one, but oriented the other way. The graphs intersect at x = -3 and 4.
So, the area is just
∫[-3,4] f(x)-g(x) dx
where
f(x) = (16-x^2)/2
g(x) = (4-x)/2
So, the area is just
∫[-3,4] f(x)-g(x) dx
where
f(x) = (16-x^2)/2
g(x) = (4-x)/2
http://www.wolframalpha.com/input/?i=plot+2y%3D16-x%5E2+%2C+x%2B2y%3D4
Find the intersection of the two graphs, you only really need the x values.
take vertical slices.
so the height of a slice is (8-x^2/2) - (2 - x/2)
= 6 - x^2/2 + x/2
take the definite integral of that from the left x to the right x of your intersection points
First, let's find the points of intersection between the two curves:
1. The equation 2y = 16 - x^2 can be rewritten as y = 8 - (1/2)x^2.
2. The equation x + 2y = 4 can be rewritten as y = (4 - x)/2.
Now, set the two equations equal to each other to find the x-coordinate at the points of intersection:
8 - (1/2)x^2 = (4 - x)/2.
Multiply both sides of the equation by 2 to get rid of the fractions:
16 - x^2 = 4 - x.
Rearrange the equation to get a quadratic equation:
x^2 - x - 12 = 0.
This equation can be factored as (x - 4)(x + 3) = 0.
So, x = 4 or x = -3.
When x = 4, substitute it into y = 8 - (1/2)x^2 to get y = 8 - (1/2)(4)^2 = 8 - 8 = 0.
When x = -3, substitute it into y = 8 - (1/2)x^2 to get y = 8 - (1/2)(-3)^2 = 8 - 4.5 = 3.5.
Therefore, the two curves intersect at the points (4, 0) and (-3, 3.5).
Now, we can find the area of the surface of the lake by integrating the difference of the two curves with respect to x between these x-interval bounds.
The region bounded by the two curves is defined by the integral:
A = ∫[from -3 to 4] [(4 - x)/2 - (8 - (1/2)x^2)] dx.
Simplifying this integral, we get:
A = ∫[from -3 to 4] (2 - (1/2)x - 8 + (1/2)x^2) dx.
Combining like terms, we have:
A = ∫[from -3 to 4] (x^2 - x - 6) dx.
Integrating, we get:
A = [((1/3)x^3 - (1/2)x^2 - 6x)] [from -3 to 4].
Evaluating this expression, we find:
A = [((1/3)(4)^3 - (1/2)(4)^2 - 6(4))] - [((1/3)(-3)^3 - (1/2)(-3)^2 - 6(-3))].
Calculating this, we find:
A = (64/3 - 8 - 24) - (-9/3 + 9/2 + 18).
Simplifying, we get:
A = 176/3 - 36 - 9/3 + 9/2 - 18.
Further simplifying, we have:
A = (176 - 108 - 9 + 27 - 108) / 6.
Calculating this, we get:
A = (178 - 108 - 108) / 6.
Finally, simplifying, we find:
A = (-38) / 6 = -19/3.
Since the area cannot be negative, it means we made an error during the calculation or interpretation. Please double-check the given curves, equations, or integration process in your question.