determine the co-ordinate of the poin of intersection of the curves y=xx and yy=8x. sketch the two curves and find the area enclosed by the two curves.

Question

determine the co-ordinate of the poin of intersection of the curves y=xx and yy=8x. sketch the two curves and find the area enclosed by the two curves.

Answer 1

y = x^2 = sqrt(8x)

x^4 = 8x
(x^3 - 8)*x = 0
x = 0 or 2

For the area between the curves, integrate
(8x)^(1/2) - x^2
from 0 to 2.

sqrt8*2^(3/2)/(3/2) - (2^3)/3
= 16/3 - 8/3 = 8/3

Answer 2

To determine the coordinates of the point of intersection of the curves y=x^2 and y^2=8x, we need to solve the two equations simultaneously.

Step 1: Solve for x in the equation y^2=8x.
- Rewrite the equation as 8x = y^2.
- Divide both sides by 8 to get x = y^2/8.

Step 2: Substitute the expression for x into the equation y=x^2.
- Replace x with y^2/8 in y=x^2 to get y = (y^2/8)^2.
- Simplify the equation to y = y^4/64.

Step 3: Rearrange the equation to solve for y.
- Multiply both sides by 64 to get 64y = y^4.
- Rewrite the equation as y^4 - 64y = 0.
- Factor out y to obtain y(y^3 - 64) = 0.
- Set each factor equal to zero: y = 0 and y^3 - 64 = 0.

Step 4: Solve for y.
- For y = 0, the x-coordinate is x = (0^2)/8 = 0.
- For y^3 - 64 = 0, we find one of the solutions is y = 4.
Substituting y=4 into the equation x = y^2/8, we get
x = (4^2)/8 = 16/8 = 2.

So, the point of intersection is at (x, y) = (2, 4).

To sketch the two curves, plot the points on a graph:

Curve y=x^2:
- Choose a few x-values (e.g., -3, -2, -1, 0, 1, 2, 3) and calculate the corresponding y-values using the equation y=x^2.
- Plot these points on the graph and connect them to form the curve.

Curve y^2=8x:
- Similarly, choose a few x-values (e.g., -3, -2, -1, 0, 1, 2, 3) and calculate the corresponding y-values using the equation y^2=8x.
- Plot these points on the graph and connect them to form the curve.

Now, to find the area enclosed by the two curves, we can integrate the function y^2 - x^2 from x=0 to x=2. The reason being, as you can see from the graphs, the lower curve is y^2 and the upper curve is x^2 within this interval.

Using integration, the area A can be found as:
A = ∫[0,2] (y^2 - x^2) dx

Simplifying the expression:
A = ∫[0,2] (8x - x^2) dx

Evaluating the integral, the area A will give you the enclosed area between the two curves.

determine the co-ordinate of the poin of intersection of the curves y=x*x and y*y=8x. sketch the two curves and find the area enclosed by the two curves.

y = x^2 = sqrt(8x)

To determine the coordinates of the point of intersection of the curves y=x^2 and y^2=8x, we need to solve the two equations simultaneously.

determine the co-ordinate of the poin of intersection of the curves y=xx and yy=8x. sketch the two curves and find the area enclosed by the two curves.