Use spherical coordinates to evaluate the triple integral x2 + y2 + z2 DV, where E
is the ball: x2 + y2 + z2 = 16.
1/S22+

4096π / 5

Step-by-step explanation:

∫∫∫ (x² + y² + z²) dV

In spherical coordinates, x² + y² + z² = r², and dV = r² sin φ dr dθ dφ.

E is the range 0 ≤ r ≤ 4, 0 ≤ φ ≤ π, 0 ≤ θ ≤ 2π.

∫₀ᵖⁱ∫₀²ᵖⁱ∫₀⁴ (r²) (r² sin φ dr dθ dφ)

∫₀ᵖⁱ∫₀²ᵖⁱ∫₀⁴ (r⁴ sin φ) dr dθ dφ

Evaluate the first integral.

∫₀ᵖⁱ∫₀²ᵖⁱ (⅕ r⁵ sin φ)|₀⁴ dθ dφ

∫₀ᵖⁱ∫₀²ᵖⁱ (¹⁰²⁴/₅ sin φ) dθ dφ

¹⁰²⁴/₅ ∫₀ᵖⁱ∫₀²ᵖⁱ (sin φ) dθ dφ

Evaluate the second integral.

¹⁰²⁴/₅ ∫₀ᵖⁱ (θ sin φ)|₀²ᵖⁱ dφ

¹⁰²⁴/₅ ∫₀ᵖⁱ (2π sin φ) dφ

²⁰⁴⁸/₅ π ∫₀ᵖⁱ sin φ dφ

Evaluate the third integral.

²⁰⁴⁸/₅ π (-cos φ)|₀ᵖⁱ

²⁰⁴⁸/₅ π (-cos π + cos 0)

²⁰⁴⁸/₅ π (1 + 1)

⁴⁰⁹⁶/₅ π

d. 24+-0.114

step-by-step explanation:

the question requires us to calculate the approximate 95% confidence interval for the mean number of ounces of ketchup per bottle in the sample given the following statistics;

sample mean = 24

standard deviation, σ = 0.4

sample size , n = 49

the confidence interval for a population mean is calculated using the formula;

sample mean ± z-score*(σ/sqrt(n))

the z-score associated with a 95% confidence interval, from the standard normal tables, is;

1.96

substituting these values into the formula;

24 ± 1.96*(0.4/7)

= 24 ± 0.112

from the alternatives given, choice d is closest to the above expression;

answer is c

step-by-step explanation:

81/12=324/x

x=48