Chapter 5: Q38P (page 248)

$\int_{z = 0}^{2} \int_{x = z}^{2} \int_{y = 8 x}^{z} dydxdz$

Short Answer

Expert verified

The solution for the given integral is $- 20$ .

Step by step solution

Definition of integration

Integration is a way of finding the whole by adding or summing the parts.

Given Information

The integral $\int_{z = 0}^{2} \int_{x = z}^{2} \int_{y = 8 x}^{z} dydxdz$ .

To find the value for the given integral.

Finding the integral value

Now, Integrate the given value,

$l = \int_{0}^{1} dz \int_{z}^{2} dx \int_{8 x}^{z} dy = \int_{0}^{2} dz \int_{z}^{2} dx (z - 8 x) = {\int_{0}^{2} dx (zx - 4 x^{2})}_{z}^{2}$

Further Integrating

Then,

$l = \int_{0}^{2} dz (2 z - z^{2} - 16 + 4 z^{2}) = \int_{0}^{2} (3 z^{2} + 2 z - 16) dz = {(z^{3} + z^{2} - 16 z)}_{0}^{2} = - 20$

Therefore, the required solution for the given integral is $- 20$ .

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$\int_{z = 0}^{2} \int_{x = z}^{2} \int_{y = 8 x}^{z} dydxdz$

Short Answer

Step by step solution

Definition of integration

Given Information

Finding the integral value

Further Integrating

One App. One Place for Learning.

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∫z=02∫x=z2∫y=8xzdydxdz

Short Answer

Step by step solution

Definition of integration

Given Information

Finding the integral value

Further Integrating

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Circular Motion and Gravitation

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Solid State Physics

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Modern Physics

Fluids

Study anywhere. Anytime. Across all devices.

$\int_{z = 0}^{2} \int_{x = z}^{2} \int_{y = 8 x}^{z} dydxdz$