Chapter 13: Q 57. (page 1016)

Sketch the region determined by the iterated integral and then evaluate the integral. For some of these integrals, it may be helpful to reverse the order of integration.
$\int_{π / 4}^{3 π / 4} \int_{0}^{\csc y} \csc y d x d y$

Short Answer

Expert verified

The value of integral is $2$

Step by step solution

Given Information

It is given that $\int_{π / 4}^{3 π / 4} \int_{0}^{\csc y} \csc y d x d y$

Simplification

The region of integration bounded by curve $y = \csc^{- 1} x$ , line $x = 0$ , $y = \frac{π}{4}$ and $y = \frac{3 π}{4}$

Integral is simplified as

$\int_{π / 4}^{3 π / 4} \int_{0}^{\csc y} \csc y d x d y = \int_{π / 4}^{3 π / 4} [x]_{0}^{ssc y} \csc y d y$

$= \int_{π / 4}^{3 π / 4} \csc^{2} y d y$

Using $\cos x = t$ , role="math" localid="1653934317274" $\Rightarrow - \csc^{2} x d x = d t$

Limits are $t = \cot \frac{π}{4} = 1$

and

$t = \cot \frac{3 π}{4} = - 1$

Hence,

$\int_{π / 4}^{3 π / 4} \int_{0}^{esc y} \csc y d x d y = - \int_{1}^{- 1} d t$

$= - [t]_{1}^{- 1}$

$= 2$

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Sketch the region determined by the iterated integral and then evaluate the integral. For some of these integrals, it may be helpful to reverse the order of integration.
$\int_{π / 4}^{3 π / 4} \int_{0}^{\csc y} \csc y d x d y$

Short Answer

Step by step solution

Given Information

Simplification

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Sketch the region determined by the iterated integral and then evaluate the integral. For some of these integrals, it may be helpful to reverse the order of integration.∫π/43π/4∫0cscycscydxdy

Short Answer

Step by step solution

Given Information

Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Sketch the region determined by the iterated integral and then evaluate the integral. For some of these integrals, it may be helpful to reverse the order of integration.
$\int_{π / 4}^{3 π / 4} \int_{0}^{\csc y} \csc y d x d y$