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Let 
R be the reglon enclosed by the line 
y=-1, the line 
x=2 and the curve 
y=x^(2)-1.
A solid is generated by rotating 
R about the line 
y=-1.
Which one of the definite integrals gives the volume of the solid?
Choose 1 answer:
(A) 
piint_(0)^(2)x^(4)dx
(B) 
piint_(-1)^(3)(x^(2)-1)^(2)dx
(c) 
piint_(-1)^(3)x^(4)dx
(D) 
piint_(0)^(2)(x^(2)-1)^(2)dx

Let R R be the reglon enclosed by the line y=1 y=-1 , the line x=2 x=2 and the curve y=x21 y=x^{2}-1 .\newlineA solid is generated by rotating R R about the line y=1 y=-1 .\newlineWhich one of the definite integrals gives the volume of the solid?\newlineChoose 11 answer:\newline(A) π02x4dx \pi \int_{0}^{2} x^{4} d x \newline(B) π13(x21)2dx \pi \int_{-1}^{3}\left(x^{2}-1\right)^{2} d x \newline(c) π13x4dx \pi \int_{-1}^{3} x^{4} d x \newline(D) π02(x21)2dx \pi \int_{0}^{2}\left(x^{2}-1\right)^{2} d x

Full solution

Q. Let R R be the reglon enclosed by the line y=1 y=-1 , the line x=2 x=2 and the curve y=x21 y=x^{2}-1 .\newlineA solid is generated by rotating R R about the line y=1 y=-1 .\newlineWhich one of the definite integrals gives the volume of the solid?\newlineChoose 11 answer:\newline(A) π02x4dx \pi \int_{0}^{2} x^{4} d x \newline(B) π13(x21)2dx \pi \int_{-1}^{3}\left(x^{2}-1\right)^{2} d x \newline(c) π13x4dx \pi \int_{-1}^{3} x^{4} d x \newline(D) π02(x21)2dx \pi \int_{0}^{2}\left(x^{2}-1\right)^{2} d x
  1. Identify Bounds of Integration: Identify the bounds of integration for the region RR. The region is bounded by y=1y = -1, x=2x = 2, and y=x21y = x^2 - 1. The bounds for xx are from 00 to 22.
  2. Formula for Volume: Determine the formula for the volume of a solid of revolution using the washer method.\newlineThe volume VV is given by V=πab(outer radius)2(inner radius)2dxV = \pi\int_{a}^{b} (\text{outer radius})^2 - (\text{inner radius})^2 \, dx.
  3. Calculate Outer Radius: Calculate the outer radius. The outer radius is the distance from the line y=1y = -1 to the curve y=x21y = x^2 - 1, which is (x21)(1)=x2(x^2 - 1) - (-1) = x^2.
  4. Calculate Inner Radius: Calculate the inner radius. The inner radius is the distance from the line y=1y = -1 to the line y=1y = -1, which is 00.
  5. Set up Integral for Volume: Set up the integral for the volume of the solid. V=π02(x2)2dx=π02x4dxV = \pi\int_{0}^{2} (x^2)^2 \,dx = \pi\int_{0}^{2} x^4 \,dx.
  6. Identify Correct Answer: Identify the correct answer from the given options.\newlineThe correct integral is π02x4dx\pi\int_{0}^{2} x^4 \, dx, which matches option (A)(A).

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