Question

Sketch the region bounded by the curves x=2y² and x=y+1 then use the shell method to find the volume of the solid generated by revolving this region about the y-axis.

Answer 1

The volume of generated by the curves and the y-axis is 0.56π unit³.

How to determine the volume generated by curves.

Given

x = 2y² and x = y + 1

Let's find the point of intersection by equating the two functions.

2y² = y + 1

2y² - y - 1 = 0

Solve the quadratic equation

Using quadratic formula

y = (-b +- √b² - 4ac)/2a

a = 2, b = -1 and c = -1

y = -(-1) +- √2² - 4(2)(-1)/2(2)

y = (1 +- √4 + 8)/4

y = (1 +- √12)/4

y = (1 + - 2√3)/4

y = 1 +- 3.5/4

= 1 + 3.5/4 or 1 - 3.5/4

= 1.11 or -0.625

To set up the integral using the shell method

\V = 2π∫]a to b{f(x) - g(x)}dy

Given x = 2y² and x = y + 1

where y ranges from -0.625 to 1.11 the integral becomes:

V = 2π∫₋₀.₆₂₅¹*¹¹(2y² - (y + 1))ydy

V = 2π∫₋₀.₆₂₅¹*¹¹(2y³ - y² - y)dy

V = 2π[y⁴/2 - y³/3 - y²/2]₋₀·₆₂₅¹*¹¹

= 2π[(1.11⁴/2 - 1.11³/3 - 1.11²/2) - (-0.625⁴/2 - (-0.625)³/3 - (-0.625)²/2]

V = 2π(0.76 - 0.46 - 0.62) -(0.08 + 0.08 -0.20)

V = 2π * 0.28

V = 0.56π unit³

The volume of generated by the curves and the y-axis is 0.56π unit³.

Answer 2

The student's question involves sketching two curves and using the shell method to compute the volume of a solid of revolution about the y-axis. It requires identifying the region, setting up the shell method integral, and evaluating the integral to find the volume.

Step-by-step explanation:

The question involves sketching the region bounded by the curves x=2y² and x=y+1, and using the shell method to find the volume of the solid generated by revolving this region about the y-axis. First, we sketch the region: x=2y² represents a parabola opening to the right, and x=y+1 represents a line

Next, we set up the shell method. We consider a typical element of thickness Δy at height y, which upon revolving about the y-axis, generates a cylindrical shell. The radius of the shell is x (since it's distance from the y-axis), and its height is the difference between the x-values of the curves: therefore, the height of the shell is (y+1) - (2y²).

The formula for the volume of a cylindrical shell is V = 2π(radius)(height)(thickness). We integrate this from the lower bound of y to the upper bound, which we find by solving the equations x=2y² and x=y+1 simultaneously for y. Substituting the expressions for the radius and the height, we get the integral ∫ 2πy((y+1)-2y²) dy to find the volume.

Lastly, we solve the integral and obtain the complete calculated answer for the volume.