The region under the curves #y=3/4x, y=1-x, y=x-1/x# is rotated about a) the x axis and b) the y axis. How do you sketch the region and find the volumes of the two solids of revolution?

Answer 1

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Answer 2

To sketch the region bounded by the curves ( y = \frac{3}{4}x ), ( y = 1 - x ), and ( y = x - \frac{1}{x} ) and find the volumes of the solids of revolution obtained by rotating this region about the x-axis and y-axis, follow these steps:

  1. Sketching the Region:

    • Plot the graphs of the three functions on the same set of axes.
    • Identify the points of intersection to determine the boundaries of the region.
  2. Finding Intersections:

    • Solve the equations of the curves pairwise to find their points of intersection.
    • These points will define the boundaries of the region.
  3. Determining Boundaries:

    • Identify the x-values at which the curves intersect to establish the boundaries for integration.
  4. Volume Calculation:

    • For rotation about the x-axis, the volume can be calculated using the formula ( V_x = \pi \int_{a}^{b} [f(x)]^2 , dx ), where ( f(x) ) represents the distance from the axis of rotation to the curve.
    • For rotation about the y-axis, use the formula ( V_y = \pi \int_{c}^{d} [g(y)]^2 , dy ), where ( g(y) ) represents the distance from the axis of rotation to the curve.
  5. Integration:

    • Integrate with respect to ( x ) or ( y ), depending on the axis of rotation, using the appropriate limits determined by the points of intersection.
  6. Evaluate the Integrals:

    • Substitute the upper and lower limits of integration into the integrals and evaluate the resulting expressions.
  7. Compute Volumes:

    • Calculate the volumes of the solids of revolution by evaluating the integrals obtained in step 6.
  8. Finalize:

    • Express the volumes with correct units and any necessary simplification.

By following these steps, you can sketch the region bounded by the given curves and determine the volumes of the solids of revolution obtained by rotating this region about the x-axis and y-axis.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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