Question

Помогите пожалуйста решить y^2=4x;y=x вычислить площадь,объем тела образованого вращением вокруг оси Ох ограничеными линиями.Сделать чертеж заранее спасибо

Answer 1

Решение во вложении-------------------

Answer 2

Problem Statement

You are given the equation y^2 = 4x and y = x + k, where k is a constant. You need to find the area and volume of the solid formed by rotating the region bounded by these curves around the x-axis.

Solution

To find the area and volume of the solid, we need to determine the limits of integration and set up the appropriate integrals.

Limits of Integration

To determine the limits of integration, we need to find the x-values where the curves intersect.

First, let's substitute y = x + k into the equation y^2 = 4x: (x + k)^2 = 4x

Expanding and rearranging the equation, we get: x^2 + 2kx + k^2 = 4x x^2 + (2k - 4)x + k^2 = 0

This is a quadratic equation in terms of x. We can solve it using the quadratic formula: x = (-b ± sqrt(b^2 - 4ac)) / (2a)

Comparing the equation to the quadratic formula, we have: a = 1 b = 2k - 4 c = k^2

Substituting these values into the quadratic formula, we get: x = (-2k + 4 ± sqrt((2k - 4)^2 - 4(k^2))) / 2

Simplifying further, we have: x = (-2k + 4 ± sqrt(4k^2 - 16k + 16 - 4k^2)) / 2 x = (-2k + 4 ± sqrt(-12k + 16)) / 2 x = (-k + 2 ± sqrt(-3k + 4))

Since we are rotating around the x-axis, the limits of integration will be the x-values where the curves intersect. Therefore, the limits of integration are x = (-k + 2 - sqrt(-3k + 4)) and x = (-k + 2 + sqrt(-3k + 4)).

Area of the Solid

To find the area of the solid, we will use the formula for the area of a solid of revolution: A = ∫[a, b] 2πy(x) √(1 + (dy/dx)^2) dx

Substituting y = x + k into the formula, we have: A = ∫[a, b] 2π(x + k) √(1 + (dy/dx)^2) dx

To find dy/dx, we differentiate y = x + k with respect to x: dy/dx = 1

Substituting this back into the formula, we have: A = ∫[a, b] 2π(x + k) √(1 + 1) dx A = ∫[a, b] 2π(x + k) √2 dx A = 2π√2 ∫[a, b] (x + k) dx A = 2π√2 [(1/2)x^2 + kx] [a, b] A = π√2 [(b^2 - a^2)/2 + 2k(b - a)]

Volume of the Solid

To find the volume of the solid, we will use the formula for the volume of a solid of revolution: V = ∫[a, b] πy(x)^2 dx

Substituting y = x + k into the formula, we have: V = ∫[a, b] π(x + k)^2 dx

Expanding and simplifying the equation, we get: V = ∫[a, b] π(x^2 + 2kx + k^2) dx V = π [(1/3)x^3 + kx^2 + k^2x] [a, b] V = π [(1/3)b^3 - (1/3)a^3 + k(b^2 - a^2) + k^2(b - a)]

Conclusion

In conclusion, the area of the solid formed by rotating the region bounded by the curves y^2 = 4x and y = x + k around the x-axis is given by the formula: A = π√2 [(b^2 - a^2)/2 + 2k(b - a)]

And the volume of the solid is given by the formula: V = π [(1/3)b^3 - (1/3)a^3 + k(b^2 - a^2) + k^2(b - a)]

Please note that the specific values of a and b will depend on the value of k. To create a diagram, you can plot the curves y^2 = 4x and y = x + k on a graphing software or use a graphing calculator.

I hope this helps! Let me know if you have any further questions.