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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.

鈭�蟺/43蟺/4鈭�0cscycscydxdy

Short Answer

Expert verified

The value of integral is2

Step by step solution

01

Given Information

It is given that 鈭�蟺/43蟺/4鈭�0cscycscydxdy

02

Simplification

The region of integration bounded by curve y=csc-1x, line x=0, y=蟺4and y=3蟺4

Integral is simplified as

鈭�蟺/43蟺/4鈭�0cscycscydxdy=鈭�蟺/43蟺/4[x]0sscycscydy

=鈭�蟺/43蟺/4csc2ydy

Using cosx=t, role="math" localid="1653934317274" 鈬�-csc2xdx=dt

Limits are t=cot蟺4=1

and

t=cot3蟺4=-1

Hence,

鈭�蟺/43蟺/4鈭�0escycscydxdy=-鈭�1-1dt

=-[t]1-1

=2

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Most popular questions from this chapter

Use the lamina from Exercise 64, but assume that the density is proportional to the distance from the x-axis.

In Exercises 57鈥�60, let R be the rectangular solid defined by

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Assume that the density of R is uniform throughout, and find the moment of inertia about the x-axis and the radius of gyration about the x-axis.

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If the density at each point in S is proportional to the point鈥檚 distance from the origin, find the center of mass of S.

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