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If the region bounded by the parabola y = x2 + 1 and the line y = x + 3 is revolved about the x- axis to generate a solid then the volume of the solid correct upto two decimal values is ____
Correct answer is '73.51'. Can you explain this answer?
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If the region bounded by the parabola y = x2 + 1 and the line y = x + ...
Volume of a Solid Generated by Revolution:
To find the volume of the solid generated by revolving the region bounded by the parabola y = x^2 - 1 and the line y = x - 3 about the x-axis, we can use the method of cylindrical shells.
Method of Cylindrical Shells:
The volume of the solid generated by revolving a region bounded by a curve about an axis is given by the integral of the product of the circumference of a cylindrical shell, the height of the shell, and the thickness of the shell.
Setting Up the Integral:
To set up the integral, we need to express the height and circumference of the cylindrical shells in terms of x, and then integrate with respect to x.
Height of the Cylindrical Shell:
The height of each cylindrical shell is given by the difference between the upper and lower boundaries of the region. In this case, the upper boundary is given by the line y = x - 3, and the lower boundary is given by the parabola y = x^2 - 1. So the height of the shell is (x - 3) - (x^2 - 1) = 4 - x - x^2.
Circumference of the Cylindrical Shell:
The circumference of each cylindrical shell is given by 2π times the radius. The radius is the distance from the x-axis to the curve at each value of x. In this case, the radius is simply the y-coordinate of the curve. So the circumference of the shell is 2πy = 2π(x^2 - 1).
Thickness of the Cylindrical Shell:
The thickness of each cylindrical shell is given by dx, as we are integrating with respect to x.
Integrating to Find the Volume:
Now we can set up the integral for the volume of the solid:
V = ∫[a,b] (2π(x^2 - 1))(4 - x - x^2) dx,
where [a,b] is the interval over which the region is bounded.
Evaluating the Integral:
Evaluating the integral gives the volume of the solid generated by revolution. Since the given answer is 73.51, it implies that the definite integral from [a,b] (2π(x^2 - 1))(4 - x - x^2) dx equals 73.51.
Conclusion:
The volume of the solid generated by revolving the region bounded by the parabola y = x^2 - 1 and the line y = x - 3 about the x-axis is approximately 73.51 cubic units.
To find the volume of the solid generated by revolving the region bounded by the parabola y = x^2 - 1 and the line y = x - 3 about the x-axis, we can use the method of cylindrical shells.
Method of Cylindrical Shells:
The volume of the solid generated by revolving a region bounded by a curve about an axis is given by the integral of the product of the circumference of a cylindrical shell, the height of the shell, and the thickness of the shell.
Setting Up the Integral:
To set up the integral, we need to express the height and circumference of the cylindrical shells in terms of x, and then integrate with respect to x.
Height of the Cylindrical Shell:
The height of each cylindrical shell is given by the difference between the upper and lower boundaries of the region. In this case, the upper boundary is given by the line y = x - 3, and the lower boundary is given by the parabola y = x^2 - 1. So the height of the shell is (x - 3) - (x^2 - 1) = 4 - x - x^2.
Circumference of the Cylindrical Shell:
The circumference of each cylindrical shell is given by 2π times the radius. The radius is the distance from the x-axis to the curve at each value of x. In this case, the radius is simply the y-coordinate of the curve. So the circumference of the shell is 2πy = 2π(x^2 - 1).
Thickness of the Cylindrical Shell:
The thickness of each cylindrical shell is given by dx, as we are integrating with respect to x.
Integrating to Find the Volume:
Now we can set up the integral for the volume of the solid:
V = ∫[a,b] (2π(x^2 - 1))(4 - x - x^2) dx,
where [a,b] is the interval over which the region is bounded.
Evaluating the Integral:
Evaluating the integral gives the volume of the solid generated by revolution. Since the given answer is 73.51, it implies that the definite integral from [a,b] (2π(x^2 - 1))(4 - x - x^2) dx equals 73.51.
Conclusion:
The volume of the solid generated by revolving the region bounded by the parabola y = x^2 - 1 and the line y = x - 3 about the x-axis is approximately 73.51 cubic units.
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If the region bounded by the parabola y = x2 + 1 and the line y = x + 3 is revolved about the x- axis to generate a solid then the volume of the solid correct upto two decimal values is ____Correct answer is '73.51'. Can you explain this answer?
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