What is the volume obtained by rotating the graph y=x^2 for 0<y<1 about the oblique axis y=x
To find the volume obtained by rotating the graph of y = x^2 for 0 < y < 1 about the oblique axis y = x, we can use the method of cylindrical shells.
First, let's consider how a typical cylindrical shell looks in this case. Take a small strip of width dx parallel to the x-axis at a distance x from the y-axis (since y = x on the oblique axis y = x). The height of this strip can be represented as y = x^2, and the length of the strip is simply dx.
To find the volume of this cylindrical shell, we multiply the height, the length, and the width:
dV = 2πxy dx
Next, we need to determine the limits of integration to cover the region where 0 < y < 1. Since y = x^2, if 0 < y < 1, then sqrt(y) < x < -sqrt(y). Therefore, the limits of integration for x are:
-x = sqrt(y) => x = -sqrt(y) (lower limit)
x = sqrt(y) (upper limit)
Finally, we integrate the expression for dV over these limits:
V = ∫ [from 0 to 1] 2πxy dx
V = 2π ∫ [from 0 to 1] x * x^2 dx
V = 2π ∫ [from 0 to 1] x^3 dx
Evaluating the integral:
V = 2π * [x^4/4] [from 0 to 1]
V = 2π * (1/4 - 0)
V = π/2
Therefore, the volume obtained by rotating the graph of y = x^2 for 0 < y < 1 about the oblique axis y = x is π/2.