Question

Find area enclosed by the given curves: x=y⁴, y=√(2-x), y=0.

Answer 1

The area enclosed by the curves x = y⁴, y = √(2 - x), and y = 0 is approximately 0.6667 square units.

Step-by-step explanation:

We first need to identify the points of intersection between the curves.

To find the intersection points between the curves x = y⁴ and y = √(2 - x), we can equate the two equations:

y⁴ = 2 - x

Rearranging the equation, we have:

x + y⁴ = 2

From the equation y = 0, we can see that one of the curves intersects the x-axis at y = 0.

Next, let's solve for the other intersection points algebraically:

x + y⁴ = 2

y⁴ = 2 - x

Taking the fourth root of both sides:

y = (2 - x)^(1/4)

Now, we can set up the definite integral to find the area enclosed between the curves:

Area = ∫[a,b] [y = √(2 - x)] - [y = y⁴] dx

To find the bounds of integration, we need to determine the x-values of the intersection points.

At y = 0, the x-value is 2, since y = 0 is equivalent to x = 2 from the equation y⁴ = 2 - x.

For the other intersection point, we equate the equations y = √(2 - x) and y = y⁴:

√(2 - x) = y⁴

Squaring both sides:

2 - x = y⁸

Rearranging the equation and substituting y = √(2 - x):

2 - x = (2 - x)⁴

Expanding the equation:

2 - x = 16 - 32x + 24x² - 8x³ + x⁴

Rearranging the terms and simplifying:

x⁴ - 8x³ + 24x² - 33x + 14 = 0

Solving this equation may require numerical methods or factoring techniques. Upon solving, we find that one of the solutions is x = 1.

So, the bounds of integration for the area integral are from x = 1 to x = 2.

The definite integral to find the area becomes:

Area = ∫[1,2] [√(2 - x) - y⁴] dx

Evaluating this integral will give us the area enclosed by the curves x = y⁴, y = √(2 - x), and y = 0 within the given interval.

=0.6667 square units.

Answer 2

To find the enclosed area, determine the points of intersection, and use the definite integral of (upper curve - lower curve) with respect to x.

Step-by-step explanation:

To find the area enclosed by the given curves, we need to determine the points of intersection first. Equate the equations of the curves to each other and solve for x:

x = y^4 and y = √(2-x)

Substitute the first equation into the second equation:

y = √(2-y^4)

Square both sides of the equation:

y^2 = 2-y^4

Rearrange the equation and solve for y:

2y^4 + y^2 - 2 = 0

Now, use the points of intersection to find the area enclosed:

The area is given by the definite integral of (upper curve - lower curve) with respect to x, from the x-coordinate of the left point of intersection to the x-coordinate of the right point of intersection:

area = ∫(y√(2-x) - y^4)dx from x = a to x = b

Once you have the values of a and b, you can evaluate the integral to find the enclosed area.