Concept explainers
a.
To write : The volume of the solid as an improper
a.
Answer to Problem 57RE
The volume of a solid as an improper integral is
Explanation of Solution
Given information :
The infinite region in the first quadrant bounded by the coordinate axes and the curve
Calculation :
Since, the region is in first quadrant and is revolved about the y-axis to generate a solid. Therefore,
Area,
Also, it is rotated about the y-axis. Thus, represent the equation in terms of y.
The radius will be x, so the area will be
The integral form 0 to infinity represents the volume. Therefore, volume
Hence,
The volume of a solid as an improper integral is
b.
To express : The integral in part (a) as a limit of a definite integral.
b.
Answer to Problem 57RE
The improper integral as a limit of a definite integral is
Explanation of Solution
Given information :
The infinite region in the first quadrant bounded by the coordinate axes and the curve
Calculation :
From part (a) volume of a solid as an improper integral is
Since, the above volume function is continuous is
Hence,
The improper integral as a limit of a definite integral is
c.
To find : The volume of the solid.
c.
Answer to Problem 57RE
The volume of the solid is
Explanation of Solution
Given information :
The infinite region in the first quadrant bounded by the coordinate axes and the curve
Calculation :
From (b) the improper integral as a limit of a definite integral is
Solve the above equation to find the volume:
Hence,
The volume of the solid is
Chapter 9 Solutions
Calculus: Graphical, Numerical, Algebraic
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