3. Siendo \( R \) limitada por \( y=\frac{1}{x^{3}}, x=1, x=3 \) e \( y=0 \), planteen las integrales con las que se calcula: i) el área de \( R \). ii) el volumen del sólido que genera \( R \) al rotar alrededor del eje \( y \). iii) el volumen del sólido que genera \( R \) al rotar alrededor de la recta \( y=-1 \). iv) el volumen del sólido que genera \( R \) al rotar alrededor de la recta \( x=4 \). 4. Calculen mediante integrales : a) el volumen de una esfera de radio \( a \). b) el volumen de un cono circular recto, siendo r el radio de la base y \( h \) la altura.

Alright, I need to address the given mathematical problems step by step. Let's start by understanding the first question, which involves setting up integrals for various geometric calculations based on the region \( R \) defined by the curves \( y = \frac{1}{x^{3}} \), \( x = 1 \), \( x = 3 \), and \( y = 0 \). **Problem 3:** - **i) Área de \( R \):** To find the area of region \( R \), I'll set up an integral with respect to \( x \) from \( x = 1 \) to \( x = 3 \). The area \( A \) can be calculated as: \[ A = \int_{1}^{3} \frac{1}{x^{3}} \, dx \] - **ii) Volumen del sólido generado por \( R \) al rotar alrededor del eje \( y \):** For the volume of the solid of revolution around the \( y \)-axis, I'll use the method of cylindrical shells. The volume \( V \) is given by: \[ V = 2\pi \int_{1}^{3} x \cdot \frac{1}{x^{3}} \, dx = 2\pi \int_{1}^{3} \frac{1}{x^{2}} \, dx \] - **iii) Volumen del sólido generado por \( R \) al rotar alrededor de la recta \( y = -1 \):** Rotating around a horizontal line requires adjusting the radius of the shell. The volume \( V \) is: \[ V = 2\pi \int_{1}^{3} (x - (-1)) \cdot \frac{1}{x^{3}} \, dx = 2\pi \int_{1}^{3} \left( x + 1 \right) \cdot \frac{1}{x^{3}} \, dx \] - **iv) Volumen del sólido generado por \( R \) al rotar alrededor de la recta \( x = 4 \):** Rotating around a vertical line involves integrating with respect to \( y \). First, express \( x \) in terms of \( y \): \[ y = \frac{1}{x^{3}} \implies x = \left( \frac{1}{y} \right)^{1/3} \] The limits for \( y \) are from \( y = 0 \) to \( y = \frac{1}{1^{3}} = 1 \). The volume \( V \) is: \[ V = \pi \int_{0}^{1} \left[ \left( 4 - \left( \frac{1}{y} \right)^{1/3} \right)^2 - \left( 4 - 1 \right)^2 \right] \, dy \] **Problem 4:** - **a) Volumen de una esfera de radio \( a \):** The volume \( V \) of a sphere is given by the integral: \[ V = \frac{4}{3} \pi a^{3} \] - **b) Volumen de un cono circular recto con radio de la base \( r \) y altura \( h \):** The volume \( V \) of a right circular cone is: \[ V = \frac{1}{3} \pi r^{2} h \] I need to ensure that all integrals are correctly set up and that the limits of integration are appropriate for each scenario. Additionally, for the sphere and cone, the standard formulas are well-known, but it's good to verify them using integration for completeness.