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Calculate the volume of the solid formed by rotating the area between

y = x^2

...

Dec 12, 2023

Calculate the volume of the solid formed by rotating the area between

y = x^2

and

y = 0

around the x-axis from

x = 0

x = 1

Solution by Steps

step 1

To calculate the volume of the solid formed by rotating the area between

y = x^2

and

y = 0

around the x-axis, we use the disk method. The volume

V

is given by the integral of

\pi y^2

with respect to

x

from

x = 0

x = 1

step 2

Setting up the integral for the volume:

V = \int_{0}^{1} \pi (x^2)^2 dx

step 3

Simplifying the integrand:

V = \pi \int_{0}^{1} x^4 dx

step 4

Calculating the integral using the power rule for integration:

V = \pi \left[ \frac{x^5}{5} \right]_{0}^{1}

step 5

Evaluating the definite integral:

V = \pi \left( \frac{1^5}{5} - \frac{0^5}{5} \right)

step 6

Simplifying the expression:

V = \frac{\pi}{5}

Answer

The volume of the solid is

\frac{\pi}{5}

cubic units.

Key Concept

Disk method for finding volumes of solids of revolution

Explanation

The volume of a solid formed by rotating a function

y = f(x)

around the x-axis from

x = a

x = b

is given by the integral of

\pi [f(x)]^2

with respect to

x

over the interval [a, b].