Calculus Homework Guidance: Riemann Sums, Integrals, Theorems, and Volumes

Q9. Slicing a Sphere: Disc Radius, Area, and Volume

Background

Topic: Applications of Integration in Geometry

This question explores how slicing a sphere with a plane creates a disc, and how to express the radius and area of that disc in terms of the sphere's radius and the position of the slice. It also asks you to sketch the area as a function of the slice position and to compute the sphere's volume using integration.

Key Terms and Formulas

• Sphere equation: $x^2 + y^2 + z^2 = R^2$

• Disc radius at $x = t$: $r = \sqrt{R^2 - t^2}$

• Disc area: $A = \pi r^2$

• Volume by integration: $V = \int_{-R}^{R} A(t) \, dt$

Step-by-Step Guidance

1. Start by considering the equation of the sphere centered at the origin: $x^2 + y^2 + z^2 = R^2$.

2. When the sphere is sliced by the plane $x = t$, the intersection forms a disc. To find the radius of this disc, set $x = t$ in the sphere's equation and solve for $y^2 + z^2$.

3. Express the radius $r$ of the disc in terms of $R$ and $t$ using $r = \sqrt{R^2 - t^2}$.

4. Write the area of the disc as a function of $t$: $A(t) = \pi (R^2 - t^2)$.

5. To sketch the area as a function of $t$, consider what happens at $t = 0$ (maximum area) and at $t = \pm R$ (area is zero). Think about the shape of the graph.

6. Set up the integral for the volume of the sphere by integrating the area of the discs from $t = -R$ to $t = R$: $V = \int_{-R}^{R} \pi (R^2 - t^2) \, dt$.

Try solving on your own before revealing the answer!

Final Answer

(1) Radius of the disc: $r = \sqrt{R^2 - t^2}$

(2) Area of the disc: $A(t) = \pi (R^2 - t^2)$

(3) Sketch: The area function $A(t)$ is a downward-opening parabola, with maximum at $t = 0$ and zero at $t = \pm R$.

(4) Volume: $V = \int_{-R}^{R} \pi (R^2 - t^2) \, dt = \frac{4}{3} \pi R^3$

This integral computes the volume of the sphere by summing up the areas of all cross-sectional discs. The result matches the well-known formula for the volume of a sphere.