# 5.2.1 vol of sin x^2 ; 0\le x \le \dfrac{\pi}{2}

• MHB
• karush
In summary, the given conversation discusses the integral $\displaystyle \int_0^{\pi/2} \sin(x^2) dx$ and how it represents area, not volume. The speaker mentions that the statement of "volume" is incomplete and asks for clarification on what type of volume is being referred to. They also mention that this integral cannot be integrated in terms of elementary functions but can be approximated to any desired accuracy.

#### karush

Gold Member
MHB
volume of the solid $y=\sin (x^2)\quad 0\le x \le \dfrac{\pi}{2}$
$\displaystyle \int_0^{\pi/2}\sin (x^2)\ dx$
ok think this should be area not volume but hope my int is set up ok

The integral you have is indeed area. Your statement of “volume” is incomplete.
Is this a volume of rotation with respect to a given axis of rotation, or is it a volume
of similar cross sections whose base lies in a region defined by the given curve,
or … something else?

If you really intend $\int_0^{\pi/2} \sin(x^2) dx$, that cannot be integrated in terms of elementary function but can be approximated to any desired accuracy.

## 1. What is the meaning of "5.2.1 vol of sin x^2 ; 0\le x \le \dfrac{\pi}{2}"?

"5.2.1 vol of sin x^2 ; 0\le x \le \dfrac{\pi}{2}" refers to the volume of a solid generated by rotating the graph of the function y = sin x^2 from 0 to π/2 radians around the x-axis.

## 2. How is the volume of the solid calculated?

The volume of the solid can be calculated using the formula V = ∫π/20 (π/2 - x^2) dx, where π/2 and 0 are the limits of integration and π/2 - x^2 is the cross-sectional area at a given x-value.

## 3. Why is the interval limited to 0 to π/2 radians?

The interval 0 to π/2 radians is chosen because the function y = sin x^2 only has positive values in this range, resulting in a solid with no negative volume.

## 4. Can this volume be represented geometrically?

Yes, the volume can be represented geometrically as a solid of revolution, where the cross-sectional areas are circles with varying radii.

## 5. How can this concept be applied in real-life scenarios?

This concept can be applied in real-life scenarios such as calculating the volume of a cylindrical container with a curved base or the volume of a water tank with a sloping base.