Consider the following.
y = x2
y = 20 − x
Maple Generated Plot
(a) Find the area of the region by integrating with respect to x.
-225/2
Incorrect: Your answer is incorrect.
(b) Find the area of the region by integrating with respect to y.
I assume you want the area enclosed by the two functions.
They intersect at (-5,25) and (4,16)
area = ∫ (20-x - x^2) dx from -5 to 4
I got 243/2
If you integrate with respect to y, you will have to do it in two parts.
To find the area of the region by integrating with respect to y, we need to determine the limits of integration.
Let's analyze the given equations:
1. y = x^2
2. y = 20 - x
To find the limits of integration, we need to see where these two curves intersect. Setting the equations equal to each other:
x^2 = 20 - x
Rearranging and setting the equation equal to zero:
x^2 + x - 20 = 0
Now we can solve this quadratic equation to find the x-values where the curves intersect.
Using factoring, we can rewrite the equation as:
(x+5)(x-4) = 0
Setting each factor equal to zero, we find:
x+5 = 0 or x-4 = 0
Solving for x, we get:
x = -5 or x = 4
So the curves intersect at x = -5 and x = 4. These values will be our limits of integration.
Now, let's proceed to find the area by integrating with respect to y.
The area can be calculated by integrating the difference of the two functions:
A = ∫[y = x^2 to y = 20 - x] (dx)
To integrate with respect to y, we need to express x in terms of y. From the first equation, we have x = ±√y.
Since we are integrating with respect to y, we need to express the limits of integration in terms of y as well. Given that we found x = -5 and x = 4 as the limits in part (a), we need to substitute them with the corresponding y-values.
Using x = √y, when x = -5, we have:
√y = -5
Squaring both sides, we get:
y = 25
Using x = √y, when x = 4, we have:
√y = 4
Squaring both sides, we get:
y = 16
So the limits of integration are y = 25 to y = 16.
Now we can calculate the area by integrating the difference of the two functions:
A = ∫[y = 25 to y = 16] (20 - x - x^2) dy
Integrating this expression will give us the area of the region.
To find the area of the region using integration with respect to y, we need to rearrange the equations and solve for x in terms of y.
First, let's rearrange the equation y = x^2.
x^2 = y
Taking the square root of both sides, we get:
x = ±√y
Next, let's rearrange the equation y = 20 - x.
x = 20 - y
Now that we have x in terms of y, we can find the limits of integration by setting the two equations equal to each other and solving for y:
√y = 20 - y
Squaring both sides, we get:
y = (20 - y)^2
Expanding and simplifying, we get:
y = 400 - 40y + y^2
Rearranging the equation, we have:
y^2 - 41y + 400 = 0
Factoring the quadratic equation, we get:
(y - 16)(y - 25) = 0
So the two values of y that determine the limits of integration are y = 16 and y = 25.
Now we can set up the integral to find the area:
A = ∫(y = 16 to 25) [x(y) dx]
Since x = ±√y, we need to take into account the positive and negative values of x.
A = 2 ∫(y = 16 to 25) [√y dy]
Evaluating the integral, we have:
A = 2 ∫(y = 16 to 25) y^(1/2) dy
Using the power rule for integration, we have:
A = 2 * (2/3)[y^(3/2)]| y = 16 to 25
A = 4/3 [25^(3/2) - 16^(3/2)]
Calculating the value, we get:
A ≈ 72.20
Therefore, the area of the region by integrating with respect to y is approximately 72.20.