Volumes of Revolution

The region under the curve $y = \sqrt{x}$ between $x = 0$ and $x = 4$ rotates around the $OX$ axis, as shown in the figure.

Set up the corresponding integral $V = \pi \int_{a}^{b} [f(x)]^2 \, dx$ and calculate the volume of the solid of revolution generated, expressing the result in terms of $\pi$.

The constant function $f(x) = 2$ between $x = 0$ and $x = 5$ rotates around the $OX$ axis.

1. a) Calculate the volume of the solid generated using the integral $V = \pi \int_{0}^{5} [f(x)]^2 \, dx$.
2. b) What geometric solid has been generated? Check the result using the formula for the volume of that solid.

The semicircle $y = \sqrt{9 - x^2}$ rotates around the $OX$ axis.

1. a) What geometric solid is generated and between what limits must you integrate?
2. b) Calculate its volume using the corresponding integral.
3. c) Check the result using the formula $V = \frac{4}{3} \pi r^3$.

The segment of the line $y = x$ between $x = 0$ and $x = 3$ rotates around the $OX$ axis generating a cone.

1. a) Calculate its volume using the integral $V = \pi \int_{0}^{3} x^2 \, dx$.
2. b) Identify the radius and height of the cone and check the result using the formula $V = \frac{1}{3} \pi r^2 h$.

Marcos calculated the volume of the solid generated by the region under the curve $y = x^2 + 1$ between $x = -1$ and $x = 1$ when rotating around the $OX$ axis as follows:

«$V = \pi \int_{-1}^{1} (x^2 + 1) \, dx = \pi \left[ \frac{x^3}{3} + x \right]_{-1}^{1} = \pi \left( \frac{1}{3} + 1 + \frac{1}{3} + 1 \right) = \frac{8\pi}{3}\ \text{u}^3$»

1. a) Identify the error in his approach and explain what it consists of.
2. b) Calculate the volume correctly.

The region enclosed between the parabola $y = x^2$ and the line $y = x$, shaded in the figure, rotates around the $OX$ axis generating a hollow solid.

1. a) Find the intersection points of both curves.
2. b) Set up the volume as the difference of two integrals (washer method) and calculate it.

The profile of a ceramic cup is modeled by the curve $y = \sqrt{x}$, with $x$ and $y$ measured in centimeters, rotating around the $OX$ axis between $x = 0$ and $x = 9$.

1. a) Calculate the interior volume of the cup in $\text{cm}^3$, rounding to the nearest integer.
2. b) Knowing that $1\ \text{cm}^3 = 1\ \text{ml}$, what is the capacity of the cup in milliliters? Does this seem like a reasonable capacity for a cup? Comment on the result.

A conical laboratory funnel is obtained by revolving the line $y = 2x$, with $0 \le x \le 2$, about the $OY$-axis, as shown in the figure (measurements in decimeters).

1. a) Express $x$ in terms of $y$ and set up the integral $V = \pi \int_{c}^{d} [g(y)]^2 \, dy$ that gives the volume, stating the limits of integration.
2. b) Calculate the volume of the funnel.
3. c) Check the result using the formula for the volume of a cone, $V = \frac{1}{3} \pi r^2 h$, identifying the radius and the height of the funnel.

The nozzle of a model aircraft engine is a part whose outer wall follows the profile of the hyperbola $y = \frac{2}{x}$ between $x = 1$ and $x = 4$ (measurements in centimeters), when rotating around the $OX$ axis.

1. a) Calculate the volume of material needed to manufacture the part, assuming it is solid between the axis and the profile.
2. b) If the material has a density of $2.5\ \text{g/cm}^3$, what is the mass of the part? Round to one decimal place of a gram.

Consider the plane region bounded by the parabola $y = x^2$, the $OX$ axis and the line $x = 2$.

1. a) Calculate the volume of the solid generated by rotating around the $OX$ axis.
2. b) Calculate the volume of the solid generated by rotating around the $OY$ axis. (Hint: subtract from the cylinder of radius $2$ and height $4$ the solid generated by the region between the curve and the $OY$ axis.)
3. c) Although the rotating region is the same, the two volumes do not coincide. Explain why with reasons.

A rugby ball is shaped like an ellipsoid of revolution: its cross-section is the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, which rotates around the $OX$ axis, as shown in the figure.

1. a) Rearrange $y^2$ in the equation of the ellipse and set up the integral giving the volume of the ellipsoid.
2. b) Prove that the volume of an ellipsoid of revolution is $V = \frac{4}{3} \pi a b^2$.
3. c) A regulation ball has a length of $30\ \text{cm}$ ($a = 15\ \text{cm}$) and a maximum diameter of $18\ \text{cm}$ ($b = 9\ \text{cm}$). Calculate its volume and express it in liters, rounding to one decimal place.

The infinite region bounded by the hyperbola $y = \frac{1}{x}$, the $OX$ axis and the line $x = 1$ rotates around the $OX$ axis, generating an infinite solid known as Torricelli's trumpet, as shown in the figure.

1. a) Calculate the volume $V(b)$ of the section of the trumpet between $x = 1$ and $x = b$.
2. b) Find $\lim_{b \to +\infty} V(b)$ and interpret the result: is the volume of the full trumpet finite?
3. c) Calculate the area of the region that has rotated, that is, $\lim_{b \to +\infty} \int_{1}^{b} \frac{1}{x} dx$. Compare with the previous part and comment on the painter's paradox: you can fill the trumpet with paint, but you cannot paint its surface.