Chapter 5: Q37P (page 248)

$\int_{x = 1}^{2} \int_{y = x}^{2 x} \int_{z = 0}^{y - x} d z d y d x$

Short Answer

Expert verified

The required integral value is $\frac{7}{6}$ .

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 $\int_{x = 1}^{2} \int_{y = x}^{2 x} \int_{z = 0}^{y - x} d z d y d x$ ,

To find the value for the given integral.

Finding the integral value

Now, integrate the given value,

$I = \int_{1}^{2} dx \int_{x}^{2 x} dy \int_{0}^{y - x} dz = \int_{1}^{2} dx \int_{x}^{2 x} dy (y - x) = \int_{1}^{2} dx {(\frac{1}{2} y^{2} - xy)}_{x}^{2 x}$

Therefore,

$I = \int_{1}^{2} (2 x^{2} - \frac{1}{2} x^{2} - 2 x^{2} + x^{2}) d x = \int_{1}^{2} \frac{1}{2} x^{2} d x = \frac{1}{6} x^{3}_{1}^{2} = \frac{7}{6}$

Therefore, the required solution for the given integral is $\frac{7}{6}$ .

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$\int_{x = 1}^{2} \int_{y = x}^{2 x} \int_{z = 0}^{y - x} d z d y d x$

Short Answer

Step by step solution

Definition of integration

Given Information

Finding the integral value

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∫x=12∫y=x2x∫z=0y-xdzdydx

Short Answer

Step by step solution

Definition of integration

Given Information

Finding the integral value

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

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Study anywhere. Anytime. Across all devices.

$\int_{x = 1}^{2} \int_{y = x}^{2 x} \int_{z = 0}^{y - x} d z d y d x$