consider the area between the graphs x+3y=1 and x+9=y^2. 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(x0= and g(x)=. Alterntaively this area can be computed as a single integral
For the life of me, I do not know what the question here is. Do you have a question about this assignment?
the question is to find the area. they are two ways either respect to x or respect to y. One way either by doing it to x or to y, your going to end up adding two integrals together to get the answer and the other one the much faster way you only need to compute to integral. My question is to show how would you do it both ways respect to x and respect to y. And when I meant by what does a= , b= , c= d=, I meant the endpoints. Sorry for the confusion. And please show all your work.
To compute the area between the graphs x+3y=1 and x+9=y^2 using two different integrals, let's start by rearranging the equations to solve for x.
For x + 3y = 1:
x = 1 - 3y
For x + 9 = y^2:
x = y^2 - 9
Now, let's find the points of intersection between the graphs. Set the two equations equal to each other:
1 - 3y = y^2 - 9
Rearrange the equation and solve for y:
y^2 + 3y - 10 = 0
This equation can be factored as:
(y - 2)(y + 5) = 0
This gives two values for y: y = 2 and y = -5.
Now, let's integrate for the two separate parts of the area.
First integral:
Let a = -5 (lower limit), b = 2 (upper limit).
Let f(x) = 1 - 3y and g(x) = y^2 - 9.
The integral for the first part of the area is:
∫[a, b] (f(x) - g(x)) dx
∫[-5, 2] [(1 - 3y) - (y^2 - 9)] dx
Simplifying the integral gives us the area for the first part.
Second integral:
Now, considering the single integral approach, we can rewrite the equations in terms of y and find the limits.
For x + 3y = 1:
x = 1 - 3y
For x + 9 = y^2:
x = y^2 - 9
The limits for the integration will be the y-values of the points of intersection, which are -5 and 2.
The single integral for the entire area is:
∫[a, b] |f(y) - g(y)| dy
where a = -5 (lower limit) and b = 2 (upper limit).
∫[-5, 2] |(1 - 3y) - (y^2 - 9)| dy
Evaluate the integral to find the area for the single integral approach.
By following these steps, you can calculate the area between the graphs using both approaches.
To compute the area between the graphs x+3y=1 and x+9=y^2, we can use integrals in two different ways.
First, let's express the equations of the curves in terms of y to set up the integration with respect to y.
Equation 1: x + 3y = 1
Rearranging, we get x = 1 - 3y.
Equation 2: x + 9 = y^2
Rearranging, we get x = y^2 - 9.
To find the limits of integration for y, we need to determine the y-values at which the two curves intersect. Setting the two equations equal to each other:
1 - 3y = y^2 - 9
Rearranging, we get y^2 + 3y - 10 = 0
Factorizing, we have (y+5)(y-2) = 0
So, y = -5 or y = 2.
Now, let's compute the sum of two integrals:
We can divide the area into two parts, from y = -5 to y = 2.
Let's denote the integral from y = a to y = b as ∫f(y)dy and the integral from y = b to y = c as ∫g(y)dy.
For the lower part of the area, from y = -5 to y = 2, we have:
∫[1 - 3y]dy, with limits of integration from -5 to 2.
For the upper part of the area, from y = -5 to y = 2, we have:
∫[y^2 - 9]dy, with limits of integration from -5 to 2.
So, the area can be computed as:
Area = ∫[1 - 3y]dy from -5 to 2 + ∫[y^2 - 9]dy from -5 to 2.
Alternatively, we can also compute the area using a single integral with respect to x.
Rearranging Equation 1, we get y = (1 - x)/3.
Rearranging Equation 2, we get y = ±√(x + 9).
To find the limits of integration for x, we need to determine the x-values at which the two curves intersect. Setting the two equations equal to each other:
(1 - x)/3 = ±√(x + 9)
Simplifying, we get (1 - x)^2 = 9(x + 9).
Expanding and rearranging, we get x^2 - 16x - 72 = 0.
Solving this quadratic equation, we find the x-values to be x = 18 or x = -4.
Now, let's compute the single integral:
∫[g(x) - f(x)]dx, with limits of integration from -4 to 18.
Here, g(x) represents the upper curve y = √(x + 9), and f(x) represents the lower curve y = (1 - x)/3.
So, the area can be computed as:
Area = ∫[√(x + 9) - (1 - x)/3]dx from -4 to 18.
By evaluating these integrals, you can obtain the numerical value of the area between the given graphs using either method.