Question

Let R be the region enclosed by the y-axis, the line y=1 and the curve y=x^3. A solid is generated by rotating R about the line y=1. What is the volume of the solid

Answer 1

Answer:

The volume of the solid is 19.$\overline{142857}$ unit³

Step-by-step explanation:

The given function is y = x³

The solid is created by revolving R about the line y = 1

We have that when y = 1, x = 1

Taking the end point as x = 2, we have the volume given by the washer method as follows;

$V = \pi \cdot \int\limits^a_b {\left( [f(x)]^2 - [g(x)]^2 \right)} \, dx$

Where;

a = 1, and b = 2, we have;

g(x) = 1

$V = \pi \cdot \int\limits^{2}_1 {\left( [x^3]^2 - [1]^2 \right)} \, dx = \pi \cdot \left[\dfrac{x^7}{7} + x \right]_1^{2} = \pi \cdot \left[\dfrac{2^7}{7} +2 -\left( \dfrac{1^7}{7} + 1\right)\right] =19\dfrac{1}{7}$

The volume of the solid, V = $19\dfrac{1}{7}$ unit³ = 19.$\overline{142857}$ unit³