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In this lesson, we will learn how to solve triple integration problems as well as applications of triple integration.

Q1:

Evaluate the triple integral οΈ οΈ οΈ π₯ π¦ π§ π₯ π¦ π§ 3 0 2 0 1 0 d d d .

Q2:

Evaluate the triple integral οΈ οΈ οΈ π§ π₯ π₯ π¦ π§ 1 0 π§ 0 π¦ 0 2 d d d .

Q3:

Find the centre of mass of the solid π = { ( π₯ , π¦ , π§ ) 0 β€ π₯ β€ 1 , 0 β€ π¦ β€ 1 , 0 β€ π§ β€ 1 β π₯ β π¦ } : with the given density function π ( π₯ , π¦ , π§ ) = 1 .

Q4:

Evaluate the triple integral οΈ οΈ οΈ 1 π§ π¦ π₯ 1 0 1 β π₯ 0 1 β π₯ β π¦ 0 d d d .

Q5:

Find the volume π of the solid bounded by the three coordinate planes and the plane 3 π₯ + 2 π¦ + 5 π§ = 6 .

Q6:

Find the volume inside the cone π§ = β π₯ + π¦ 2 2 , where 0 β€ π§ β€ 3 .

Q7:

Find the centre of mass of the solid π = ο© ( π₯ , π¦ , π§ ) π₯ β₯ 0 , π¦ β₯ 0 , π§ β₯ 0 , π₯ + π¦ + π§ β€ π ο΅ : 2 2 2 2 with the given density function π ( π₯ , π¦ , π§ ) = 1 .

Q8:

Evaluate the triple integral οΈ οΈ οΈ π§ π₯ π₯ π§ π¦ π 1 π¦ 0 0 2 1 π¦ d d d .

Q9:

Evaluate the triple integral οΈ οΈ οΈ π§ π π₯ π¦ π§ 1 0 π§ 0 π¦ 0 π¦ 2 d d d .

Q10:

Evaluate the triple integral οΈ οΈ οΈ 1 π₯ π¦ π§ 2 1 4 2 3 0 d d d .

Q11:

Find the centre of mass of the solid π = { ( π₯ , π¦ , π§ ) 0 β€ π₯ β€ 1 , 0 β€ π¦ β€ 1 , 0 β€ π§ β€ 1 } : with the given density function π ( π₯ , π¦ , π§ ) = π₯ π¦ π§ .

Q12:

Find the volume inside the cone π§ = π₯ + π¦ 2 2 , where 0 β€ π§ β€ 4 .

Q13:

Find the volume inside the elliptic cylinder π₯ π + π¦ π = 1 2 2 2 2 , where 0 β€ π§ β€ 2 .

Q14:

Find, in terms of π , the volume of the region that lies within both the sphere with equation π₯ + π¦ + π§ = 4 2 2 2 and the cylinder with equation π₯ + π¦ = 1 2 2 .

Q15:

Find the volume inside both the sphere π₯ + π¦ + π§ = 1 2 2 2 and the cone π§ = β π₯ + π¦ 2 2 .

Q16:

Find the centre of mass of the solid π = { ( π₯ , π¦ , π§ ) 0 β€ π₯ β€ 1 , 0 β€ π¦ β€ 1 , 0 β€ π§ β€ 1 } : with the given density function π ( π₯ , π¦ , π§ ) = π₯ + π¦ + π§ 2 2 2 .

Q17:

Let π , π , and π be real numbers selected randomly from the interval ] 0 , 1 [ . What is the probability that the equation π π₯ + π π₯ + π = 0 2 has at least one real solution for π₯ ? Rounding the value to four decimal places.

Q18:

Evaluate the triple integral οΈ οΈ οΈ π₯ π¦ π§ π§ π¦ π₯ 1 0 π₯ 0 π¦ 0 d d d .

Q19:

Find the centre of mass of the solid π = ο© ( π₯ , π¦ , π§ ) π§ β₯ 0 , π₯ + π¦ + π§ β€ π ο΅ : 2 2 2 2 with the given density function π ( π₯ , π¦ , π§ ) = π₯ + π¦ + π§ 2 2 2 .

Q20:

Evaluate the triple integral οΈ οΈ οΈ π¦ π§ π₯ π§ π¦ 2 1 π¦ 0 π§ 0 2 2 d d d .

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