The region under the curves #y=3/4x, y=1x, y=x1/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?
See the answer below:
By signing up, you agree to our Terms of Service and Privacy Policy
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 xaxis and yaxis, follow these steps:

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.

Finding Intersections:
 Solve the equations of the curves pairwise to find their points of intersection.
 These points will define the boundaries of the region.

Determining Boundaries:
 Identify the xvalues at which the curves intersect to establish the boundaries for integration.

Volume Calculation:
 For rotation about the xaxis, 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 yaxis, 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.

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

Evaluate the Integrals:
 Substitute the upper and lower limits of integration into the integrals and evaluate the resulting expressions.

Compute Volumes:
 Calculate the volumes of the solids of revolution by evaluating the integrals obtained in step 6.

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 xaxis and yaxis.
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided 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 onesided 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 onesided 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 onesided 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.
 Let R be the region in the first quadrant enclosed by the lines #x=ln 3# and #y=1# and the graph of #y=e^(x/2)#, how do you find the volume of the solid generated when R is revolved about the line y=1?
 How do you find the area between #g(x)=4/(2x), y=4, x=0#?
 How do you find the area of the region bounded by the curves #y=x# and #y=x^22# ?
 The region under the curve #y=sqrtx# bounded by #0<=x<=4# 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?
 How are certain formulæ for areas of circles and ellipses related to calculus?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7