find volume using disc method
y=x^2,y=0,x=0,x=5
To find the volume using the disk method, follow these steps:
1. Determine the axis of revolution: In this case, the axis of revolution is the x-axis because we are rotating the region between the curves y = x^2 and y = 0 about the x-axis.
2. Set up the integral: The formula for volume using the disk method is V = π∫(R(x))^2 dx, where R(x) is the distance between the axis of revolution and the curve.
In this case, R(x) is the distance between the x-axis and the curve y = x^2, so R(x) = x^2.
3. Determine the limits of integration: We need to find the bounds for x over which the region is being rotated. In this case, the region is bounded by x = 0 and x = 5.
4. Evaluate the integral: The integral becomes V = π∫(x^2)^2 dx, with the limits of integration from x = 0 to x = 5. Simplifying the integral, we get V = π∫(x^4) dx.
Integrating x^4 with respect to x gives us V = π(x^5/5) evaluated from x = 0 to x = 5.
Evaluating the integral, we get V = π * (5^5/5 - 0^5/5).
Simplifying further, V = π * (3125/5).
Therefore, the volume using the disk method is V = 625π cubic units.