washer method!?
Find the volume of y=x^2+3, y=3, between [0,2] around y=3
make a sketch
wouldn't your volume be the same if you rotated y = x^2 around the x-axis, for [0,2] ??
I am sure you can do that easy one.
let me know what you got.
No it wouldnt be the same because one is a disk rule one is a washer rule
you are rotating it around y=3, so there is no hole.
your right im sorry, i was thinking we were rotating around x=3 lol but will you do it so i can check my answer?
I got 32pi/5
i think u forgot to add the 2x^3 to the antiderivitive for the fundamental theorm step
To find the volume using the washer method, we need to integrate the area of the cross-sections formed by rotating the region between the two curves around the given axis, which is y=3 in this case.
Step 1: Determine the outer and inner functions.
The outer function is the upper curve, which is y=3.
The inner function is the lower curve, which is y=x^2+3.
Step 2: Sketch the region and the axis of rotation.
In this case, we are rotating the region between y=x^2+3 and y=3 around the line y=3. The region is a parabolic shape.
Step 3: Find the limits of integration.
We are given that the integration limits are from x=0 to x=2.
Step 4: Set up the integral.
The formula for the volume using the washer method is V = ∫[a,b] (π((R(x))^2 - (r(x))^2)) dx, where R(x) is the outer radius and r(x) is the inner radius.
In this case, the outer radius (R(x)) is the distance between the outer function (y=3) and the axis of rotation (y=3), which is zero.
The inner radius (r(x)) is the distance between the inner function (y=x^2+3) and the axis of rotation (y=3), which is (3 - (x^2+3)) = -x^2.
Therefore, the integral is V = ∫[0,2] π((0)^2 - (-x^2)^2) dx.
Step 5: Evaluate the integral.
V = ∫[0,2] π(x^4) dx.
Integrating x^4, we get V = π * (1/5) * x^5, evaluated from 0 to 2.
V = π * (1/5) * (2^5 - 0^5)
V = π * (1/5) * 32
V = (32π)/5
The volume of the region using the washer method between y=x^2+3, y=3, and around y=3 is (32π)/5 cubic units.