Chapter 5: Q43P (page 248)

Find the volume between the surfaces $z = 2 x^{2} + y^{2} + 12$ and $z = x^{2} + y^{2} + 8$ , and over the triangle with vertices (0,0), (1,0) and (1,2).

Short Answer

Expert verified

The volume obtained for the planes over the vertices is $\frac{9}{2}$

Step by step solution

Definition of double integral and mass formula

The double integral of f(x,y) over the area A in the (x,y) plane as the limit of this sum, and write it as $\int \int_{A} f (x, y) dxdy$

Draw the area bounded by the curve in the plane

Draw the area bounded by region for the integration in the xy plane.

Calculation of the volume under the curve

Calculate the volume between the surfaces $z = 2 x^{2} + y^{2} + 12$ and $z = x^{2} + y^{2} + 8$ , and over the triangle with vertices (0,0) , (1,0) and (1,2).

$l = \int_{0}^{1} d x \int_{0}^{2 x} d y \int_{x^{2} + y^{2} + 8}^{2 x^{2} + y^{2} + 12} d z = \int_{0}^{1} d x \int_{0}^{2 x} (- x^{2} - y^{2} - 8 + 2 x^{2} + y^{2} + 12) d y = \int_{0}^{1} d x \int_{0}^{2 x} (x^{2} + 4) d y = \int_{0}^{1} d x (x^{2} + 4) 2 x$

Simplify and solve further.

$\int_{0}^{1} d x (x^{2} + 4) 2 x = \int_{0}^{1} (2 x^{3} + 8 x) d x = {[\frac{1}{2} x^{4} + 4 x^{2}]}_{0}^{1} = (\frac{1}{2} + 4) = \frac{9}{2}$

Therefore, the value is $\frac{9}{2}$ ,

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Find the volume between the surfaces $z = 2 x^{2} + y^{2} + 12$ and $z = x^{2} + y^{2} + 8$ , and over the triangle with vertices (0,0), (1,0) and (1,2).

Short Answer

Step by step solution

Definition of double integral and mass formula

Draw the area bounded by the curve in the plane

Calculation of the volume under the curve

One App. One Place for Learning.

Most popular questions from this chapter

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Find the volume between the surfaces z=2x2+y2+12and z=x2+y2+8, and over the triangle with vertices (0,0), (1,0) and (1,2).

Short Answer

Step by step solution

Definition of double integral and mass formula

Draw the area bounded by the curve in the plane

Calculation of the volume under the curve

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electrostatics

Fundamentals of Physics

Rotational Dynamics

Energy Physics

Electromagnetism

Scientific Method Physics

Study anywhere. Anytime. Across all devices.

Find the volume between the surfaces $z = 2 x^{2} + y^{2} + 12$ and $z = x^{2} + y^{2} + 8$ , and over the triangle with vertices (0,0), (1,0) and (1,2).