Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

0. Functions

Introduction to Functions

Calculus

0. Functions

Introduction to Functions

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Multiple Choice

Evaluate the triple integral of $1$ over the part of the ball defined by $x^{2} + y^{2} + z^{2} \leq 8$ that lies in the first octant.

\frac{4}{3} π

2 π

\frac{π}{2}

1

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Verified step by step guidance

Step 1: Recognize that the problem involves evaluating a triple integral over a region defined by the inequality x² + y² + z² ≤ 8, which represents a ball of radius √8 centered at the origin. The region is restricted to the first octant, where x, y, and z are all non-negative.

Step 2: Convert the integral into spherical coordinates for simplicity. In spherical coordinates, x = ρsin(φ)cos(θ), y = ρsin(φ)sin(θ), and z = ρcos(φ), where ρ is the radial distance, φ is the polar angle, and θ is the azimuthal angle. The volume element dV becomes ρ²sin(φ)dρdφdθ.

Step 3: Set up the limits of integration. In the first octant, θ ranges from 0 to π/2, φ ranges from 0 to π/2, and ρ ranges from 0 to √8 (the radius of the ball). The integrand is 1, as specified in the problem.

Step 4: Write the integral in spherical coordinates: ∫∫∫ ρ²sin(φ)dρdφdθ, with the limits of integration being ρ ∈ [0, √8], φ ∈ [0, π/2], and θ ∈ [0, π/2].

Step 5: Evaluate the integral step by step. First, integrate with respect to ρ, then φ, and finally θ, applying the limits of integration at each step. This will yield the volume of the region in the first octant.