I am having so many problems with these volume problems,
1. Find the volume of a the solid obtained by rotating the region enclosed by: x = 2y and y^3 = x with y>0 about the x axis
2.Find the volume of cone of height h = 250 and a circular base with radius r = 150 First find the linear equation r = Ah + B relating r, the radius of a cross-section of the cone, to h, the height of the cross-section: r = _h + _
Now compute the volume of revolution with respect to this equation to get the volume of the cone:
3.You are an Egyptian pharoah. You decide to honor yourself with a pyramid of height h = 3250 and a square base with side s = 1910
To impress your Egyptian subjects, find the volume of the pyramid.
First write down the integrand
Now give the volume
1. To find the volume of the solid obtained by rotating the region enclosed by the curves x = 2y and y^3 = x about the x-axis, you can use the method of cylindrical shells.
First, find the vertical distance between the curves at any given x-value. To do this, solve the two equations x = 2y and y^3 = x simultaneously. Substitute x = 2y into the second equation to get y^3 = 2y. Rearrange this equation to y(y^2 - 2) = 0, which gives y = 0 as one solution and y = ∛2 as the other solution.
Next, determine the limits of integration by finding the x-values where the two curves intersect. Set x = 2y equal to y^3 and solve for y: 2y = y^3. Rearrange this equation to y(y^2 - 2) = 0, which gives y = 0 as one solution and y = ∛2 as the other solution.
Now, set up the integral to find the volume: ∫[a, b] 2πy * (x2 - x1) dx, where a and b are the x-values where the curves intersect (∛2 and 8) and the x2 - x1 is the vertical distance between the curves at each x-value (2y - y^3).
The integrand is 2πy * (2y - y^3) and the limits of integration are from ∛2 to 8.
Evaluate this integral to find the volume of the solid.
2. To find the volume of a cone with height h = 250 and radius r = 150, we can use the formula for the volume of a cone: V = (1/3)πr^2h.
However, before calculating the volume, we need to find the linear equation relating r and h. We are given that r = Ah + B, where A and B are constants. To find A and B, we can use the given values of r and h.
Substitute r = 150 and h = 250 into the equation r = Ah + B:
150 = A(250) + B.
Solve for A and B by simplifying the equation:
A = (150 - B)/250.
Now that we have the linear equation r = Ah + B, we can proceed to calculate the volume of revolution with respect to this equation.
The volume can be calculated by using the formula V = ∫[0,h] π(r(h))^2 dx, where r(h) is the radius as a function of h, given by the equation r = Ah + B.
Substitute the linear equation r = Ah + B into the formula and simplify:
V = ∫[0,h] π((Ah + B)^2) dx.
Evaluate this integral to find the volume of the cone.
3. To find the volume of a pyramid with a height h = 3250 and a square base with side s = 1910, we can use the formula for the volume of a pyramid: V = (1/3)Bh, where B is the area of the base and h is the height.
In this case, the base is a square with side s = 1910, so the area of the base is B = s^2.
Substitute the values of s = 1910 and h = 3250 into the formula V = (1/3)Bh:
V = (1/3)(1910^2)(3250).
Calculate this expression to find the volume of the pyramid.