Exercises: 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 $V = \frac{1}{3} \pi r^2 h$.

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.

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.

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.

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.

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.

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.