# Solids of revolution, y axis

• James889
In summary, the conversation discusses finding the volume of a solid created by rotating the area e^x-1 about the y-axis, bounded by y=1, x=0, and x=ln2. The conversation also includes a correction to a mistake in the shell method formula and a clarification of the correct answer, which is 2\pi(ln2-1)^2.

#### James889

Hi,
The area
$$e^x-1$$
Is rotated about the y axis, bounded by y=1, x=0 and x=ln2 find the volume of the solid.

And i am clearly making something wrong, so if anyone could verify my work.

$$~ 2\pi\int_0^{ln2}x\cdot(1-(e^x-1)$$
$$-2\pi\int_0^{ln2}xe^x-2x$$

Integration:
u=x, du=1
dv=e^x, v=e^x

$$\int xe^x -2x= xe^x -e^x -x^2$$
$$e^x(x-1)-x^2\bigg|_0^{ln2} = 2\pi \cdot ln2-1 -(ln2)^2$$

The answer is supposed to be $$2\pi(ln2-1)^2$$
Thanks!

where did that 1 come from in your shell method formula?

When you integrate you should obtain : 2*pi*(ln22-2*ln2+1)

That then can be the factored to the book's answer.

You lost a 2 with the ln2

## 1. What is a solid of revolution?

A solid of revolution is a three-dimensional shape created by rotating a two-dimensional shape around an axis. This axis can be any line, but for the specific case of solids of revolution around the y-axis, the shape is rotated around a vertical line passing through the origin.

## 2. How are solids of revolution, y axis, used in real life?

Solids of revolution, y axis, have many practical applications in real life. For example, they are used in engineering and manufacturing to create objects with circular or symmetrical shapes, such as pipes, cylinders, and cones. They are also used in mathematics to solve problems involving volume and surface area.

## 3. What are the steps to calculate the volume of a solid of revolution, y axis?

To calculate the volume of a solid of revolution, y axis, the following steps can be followed:

1. Determine the function that represents the cross-section of the solid.
2. Find the limits of integration by identifying the points where the function intersects the y-axis.
3. Write the integral for the volume using the formula V = π∫ab[f(y)]2dy, where a and b are the limits of integration.
4. Solve the integral and simplify to find the volume of the solid.

## 4. Can solids of revolution, y axis, have holes?

Yes, solids of revolution, y axis, can have holes. These holes can be created by subtracting a smaller shape from the original shape before rotating it around the axis. This can be useful in creating objects such as hollow cylinders or donut-shaped objects.

## 5. How do you find the surface area of a solid of revolution, y axis?

To find the surface area of a solid of revolution, y axis, the following steps can be followed:

1. Determine the function that represents the cross-section of the solid.
2. Find the limits of integration by identifying the points where the function intersects the y-axis.
3. Write the integral for the surface area using the formula S = 2π∫ab[f(y)]√(1 + [f'(y)]2)dy, where a and b are the limits of integration.
4. Solve the integral and simplify to find the surface area of the solid.