224k views
2 votes
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.

2 Answers

3 votes

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.

Sketch the region bounded by the curves x=2y² and x=y+1 then use the shell method-example-1
User Whoplisp
by
8.0k points
5 votes

Final answer:

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.

User Sylph
by
7.3k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories