1. Find the volume formed by rotating the region enclosed by x=5y and x=y^3 with y¡Ý0 about the y-axis.
2. Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis y=18x-6x^2 , y=0 : about the y-axis.
PLease can anyone help me find the volume plz show ur works if u can so i can understand it better.
are you more comfortable finding volume by rotating about the x-axis.
Here is a good trick that works well in the first case.
Interchange the x and y variables in your equations and rotate about the x-axis, you get the same shape. (Basically I took the inverse)
BTW, I don't know what "..with y¡Ý0.." means.
Sure, I can help you find the volumes of the solids obtained by rotating the given regions about the y-axis. Let's start with the first question.
Question 1: Find the volume formed by rotating the region enclosed by x=5y and x=y^3 with y≥0 about the y-axis.
To find the volume, we can use the method of cylindrical shells. The formula for finding the volume using cylindrical shells is:
V = ∫ 2πx * h * dx
where x is the variable of integration, h represents the height of each cylindrical shell, and the integral is taken over the range of x values that define the region.
Step 1: Interchange the x and y variables.
In this case, we need to interchange the x and y variables. The equation x=5y becomes y=5x, and x=y^3 becomes y=x^3.
Step 2: Determine the limits of integration.
To determine the limits of integration, we need to find the x-values at which the two curves intersect. Setting y=5x and y=x^3 equal to each other, we can solve for x:
5x = x^3
x^3 - 5x = 0
x(x^2 - 5) = 0
The solutions are x=0, x=sqrt(5), and x=-sqrt(5).
So, the limits of integration are x=0 to x=sqrt(5).
Step 3: Determine the height of each cylindrical shell.
The height of each cylindrical shell is given by the difference between the two functions: h = y_upper - y_lower (where y_upper is the top curve and y_lower is the bottom curve).
In this case, the top curve is y=5x and the bottom curve is y=x^3. So, the height is h = 5x - x^3.
Step 4: Evaluate the integral.
Now we can set up and evaluate the integral to find the volume:
V = ∫ 2πx * (5x - x^3) * dx
V = 2π∫ (5x^2 - x^4) * dx
Integrating the terms of the polynomial, we get:
V = 2π[ (5/3)x^3 - (1/5)x^5 ] evaluated from x=0 to x=sqrt(5)
Evaluating the integral, we obtain:
V = 2π[ (5/3)(sqrt(5))^3 - (1/5)(sqrt(5))^5 ]
Finally, we can simplify the expression and calculate the volume.
This completes the calculation of the volume for the first question.
Now let's move on to the second question.