Exam 15: Multiple Integrals

A cylindrical drill with radius $3$ is used to bore a hole through the center of a sphere of radius $9$ . Find the volume of the ring-shaped solid that remains. Round the answer to the nearest hundredth.

(Short Answer)

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Question 121

Use polar coordinates to find the volume of the solid bounded by the paraboloid $z = 7 - 6 x ^ { 2 } - 6 y ^ { 2 }$ and the plane $z = 1$ .

(Multiple Choice)

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Question 122

Calculate the double integral. Round your answer to two decimal places. $\iint _ { R } x \sin ( x + y ) d A , \quad R = \left[ 0 , \frac { \pi } { 4 } \right] \times \left[ 0 , \frac { \pi } { 6 } \right]$

(Short Answer)

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Question 123

Find the volume under $z = 3 x ^ { 5 } + 3 y ^ { 5 }$ and above the region bounded by $y = x ^ { 2 }$ and $x = y ^ { 2 }$ .

(Multiple Choice)

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Question 124

A cylindrical drill with radius $3$ is used to bore a hole through the center of a sphere of radius $9$ . Find the volume of the ring-shaped solid that remains. Round the answer to the nearest hundredth.

Use polar coordinates to find the volume of the solid bounded by the paraboloid $z = 7 - 6 x ^ { 2 } - 6 y ^ { 2 }$ and the plane $z = 1$ .

Calculate the double integral. Round your answer to two decimal places. $\iint _ { R } x \sin ( x + y ) d A , \quad R = \left[ 0 , \frac { \pi } { 4 } \right] \times \left[ 0 , \frac { \pi } { 6 } \right]$

Find the volume under $z = 3 x ^ { 5 } + 3 y ^ { 5 }$ and above the region bounded by $y = x ^ { 2 }$ and $x = y ^ { 2 }$ .

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Exam 15: Multiple Integrals

A cylindrical drill with radius 333 is used to bore a hole through the center of a sphere of radius 999 . Find the volume of the ring-shaped solid that remains. Round the answer to the nearest hundredth.

Use polar coordinates to find the volume of the solid bounded by the paraboloid z=7−6x2−6y2z = 7 - 6 x ^ { 2 } - 6 y ^ { 2 }z=7−6x2−6y2 and the plane z=1z = 1z=1 .

Calculate the double integral. Round your answer to two decimal places. ∬Rxsin⁡(x+y)dA,R=[0,π4]×[0,π6]\iint _ { R } x \sin ( x + y ) d A , \quad R = \left[ 0 , \frac { \pi } { 4 } \right] \times \left[ 0 , \frac { \pi } { 6 } \right]∬R​xsin(x+y)dA,R=[0,4π​]×[0,6π​]

Find the volume under z=3x5+3y5z = 3 x ^ { 5 } + 3 y ^ { 5 }z=3x5+3y5 and above the region bounded by y=x2y = x ^ { 2 }y=x2 and x=y2x = y ^ { 2 }x=y2 .

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

A cylindrical drill with radius $3$ is used to bore a hole through the center of a sphere of radius $9$ . Find the volume of the ring-shaped solid that remains. Round the answer to the nearest hundredth.

Use polar coordinates to find the volume of the solid bounded by the paraboloid $z = 7 - 6 x ^ { 2 } - 6 y ^ { 2 }$ and the plane $z = 1$ .

Calculate the double integral. Round your answer to two decimal places. $\iint _ { R } x \sin ( x + y ) d A , \quad R = \left[ 0 , \frac { \pi } { 4 } \right] \times \left[ 0 , \frac { \pi } { 6 } \right]$

Find the volume under $z = 3 x ^ { 5 } + 3 y ^ { 5 }$ and above the region bounded by $y = x ^ { 2 }$ and $x = y ^ { 2 }$ .