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Compute the integral

`\int -x^(2)csc^(2)(x^(3)+1)dx`

Find the area of the region bounded by the curves

`y=\sqrt(x+1),y=(1)/(x+1)`

and

`x=4`

. Consider the region bounded by

`y=(1)/(4)x^(2)`

and

`y=(1)/(2)x`

(a) Use cylindrical shells to find the volume of the solid generated by rotating the above region about the y-axis. (b) Use Washer method to find the volume of the solid generated by rotating the above region about the

`x_()`

-axis. Determine if the following are a type 1 or type 2 improper integral. If the integral converges find the value, otherwise say it diverges. (a)

`\int_3^(\infty ) (1)/((x-2)^((3)/(2)))dx`

(b)

`\int_0^5 (1)/(\root(3)(5-x))dx`