Chapter 13: Q. 38 (page 1015)

In Exercises 35–40, find the volume of the solid bounded above by the given function over the specified region $Ω = {(x, y) | 0 \leq x \leq \frac{π}{4}$ and $\sin x \leq y \leq \cos x}$ $f (x, y) = x^{2} y .$

Short Answer

Expert verified

Volume bounded by given function is: $\frac{π^{2} - 8}{64}$

Step by step solution

Step 1. Given Information

Function, $f (x, y) = x^{2} y$

Region, $Ω = {(x, y) | 0 \leq x \leq \frac{π}{4}$ and $\sin x \leq y \leq \cos x}$

Step 2. Calculating the volume of the solid bounded above by the given function and region

The double integral is given by,

\int_{0}^{\frac{π}{4}} \int_{s i n x}^{c o s x} f (x, y) d y d x = \int_{0}^{\frac{π}{4}} \int_{s i n x}^{c o s x} x^{2} y d y d x = \int_{0}^{\frac{π}{4}} [{\frac{x^{2} y^{2}}{2}}_{\sin x}^{\cos x}] dx = \int_{0}^{\frac{π}{4}} [\frac{x^{2} (\cos^{2} x - \sin^{2} x)}{2}] dx = \int_{0}^{\frac{π}{4}} [\frac{x^{2} (\cos 2 x)}{2}] dx

Integrating by parts, we get,

${(\frac{x^{2} \sin 2 x}{4} + \frac{x \cos 2 x}{4} - \frac{\sin 2 x}{8})}_{0}^{\frac{π}{4}} = \frac{π^{2}}{64} + 0 - \frac{1}{8} - 0 + 0 - 0 = \frac{π^{2} - 8}{64}$

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In Exercises 35–40, find the volume of the solid bounded above by the given function over the specified region $Ω = {(x, y) | 0 \leq x \leq \frac{π}{4}$ and $\sin x \leq y \leq \cos x}$ $f (x, y) = x^{2} y .$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculating the volume of the solid bounded above by the given function and region

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In Exercises 35–40, find the volume of the solid bounded above by the given function over the specified regionΩ={x,y|0≤x≤π4andsinx≤y≤cosx}f(x,y)=x2y.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculating the volume of the solid bounded above by the given function and region

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Applied Mathematics

Probability and Statistics

Pure Maths

Statistics

Geometry

Study anywhere. Anytime. Across all devices.

In Exercises 35–40, find the volume of the solid bounded above by the given function over the specified region $Ω = {(x, y) | 0 \leq x \leq \frac{π}{4}$ and $\sin x \leq y \leq \cos x}$ $f (x, y) = x^{2} y .$