Consider the area between the graphs x+6y=8 and x+8=y2. This area can be computed in two different ways using integrals

First of all it can be computed as a sum of two integrals

where a= , b=, c= and
f(x)=
g(x)=

I found a, but not b or c. I can't seem to figure out f(x) and g(x). I would assume f(x) is (8-x)/6 and g(x) is (x+8)^(1/2), but it's not right.. Can you help me?

To find the values of a, b, c, f(x), and g(x) for the given area between the graphs x+6y=8 and x+8=y^2, we need to first identify the points of intersection and set up the appropriate integrals.

To find the points of intersection, we can set the two equations equal to each other:

x + 6y = 8
x + 8 = y^2

Solving these equations simultaneously, we get:

x + 6y = 8
x = y^2 - 8

Setting them equal, we have:

y^2 - 8 = 6y
y^2 - 6y - 8 = 0

Using the quadratic formula, we can solve for y:

y = (6 ± √(6^2 - 4(-8))) / 2
y = (6 ± √(36 + 32)) / 2
y = (6 ± √68) / 2
y = (6 ± 2√17) / 2
y = 3 ± √17

So we have two values for y: y = 3 + √17 and y = 3 - √17.

Now we need to find the corresponding values of x at these points. We can substitute the values of y back into either equation. Let's use the equation x = y^2 - 8:

For y = 3 + √17:
x = (3 + √17)^2 - 8

Expanding and simplifying:
x = 9 + 6√17 + 17 - 8
x = 18 + 6√17

Similarly, for y = 3 - √17:
x = (3 - √17)^2 - 8
x = 9 - 6√17 + 17 - 8
x = 18 - 6√17

So the two points of intersection are (18 + 6√17, 3 + √17) and (18 - 6√17, 3 - √17).

Now we can set up the integral to compute the area between the graphs. We need to determine the limits of integration for x and y.

For the x direction, we integrate from x = a (18 - 6√17) to x = b (18 + 6√17).

For the y direction, we integrate from y = c (3 - √17) to y = f(x).

To find the limits for f(x), we need to solve the equation x + 8 = y^2 for y:

x + 8 = y^2
y^2 = x + 8
y = √(x + 8)

So the limits for y are y = c (3 - √17) to y = √(x + 8).

Putting it all together, the area can be computed as the sum of two integrals:

A = ∫[a to b] [(√(x + 8) - (3 - √17)] dx

To evaluate this integral, you can perform the calculation from x = a to x = b and substitute the values of a and b that we found earlier.

To find f(x) and g(x), we need to rewrite the equations x+6y=8 and x+8=y^2 in terms of y.

Starting with the equation x+6y=8, let's solve it for y:
x + 6y = 8
6y = 8 - x
y = (8 - x)/6

So, f(x) is (8 - x)/6.

Now let's rewrite the equation x+8=y^2 in terms of y:
x + 8 = y^2

To solve for y, we'll take the square root of both sides:
√(x + 8) = y

So, g(x) is √(x + 8).

Therefore, the area between the graphs can be computed as a sum of two integrals:

∫[a,b] [f(x) - g(x)] dx + ∫[b,c] [g(x) - f(x)] dx

where a, b, and c are the x-values where the graphs intersect. You mentioned that you found a, so you can substitute that value into the integral. However, you still need to find the x-values b and c, which are the points of intersection between the two graphs.

Here is a sketch of the region

http://www.wolframalpha.com/input/?i=solve+x%2B6y%3D8+%2C+x%2B8%3Dy%5E2

I will do it the easiest way.

The intersection points are (-4,2) and (56,-8)

I would take horizontal slices
effective width of a horizontal slice
= 8-6y - (y^2 - 8
= 16 - 6y - y^2

Area = ∫(16 - 6y - y^2) dy from y=-8 to 2
= (16y - 3y^2 - (1/3)y^3) | from -8 to 2
= (32 - 12 - 8/3) - (-128 - 192 + 512/3)
= 500/3 or
166 2/3

check my arithmetic