Let R be the shaded region in the first quadrant enclosed by the y-axis and the graphs of #y=sin(x)# and #y=cos(x)#, how do you find the volume of the solid whose base is R and whose cross sections, cut by planes perpendicular to the x-axis, are squares?

Answer 1

See the answer below:

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the volume of the solid whose base is the shaded region ( R ) and whose cross sections, cut by planes perpendicular to the x-axis, are squares, you would integrate the area of the squares with respect to ( x ).

Since the squares are perpendicular to the x-axis, their side length would be equal to the difference between the y-values of the curves ( y = \sin(x) ) and ( y = \cos(x) ).

Thus, the side length of each square would be ( \sin(x) - \cos(x) ).

To find the volume, integrate the square of the side length over the interval where the curves intersect, which is ([0, \pi/4]):

[ V = \int_{0}^{\pi/4} (\sin(x) - \cos(x))^2 , dx ]

Solve this integral to find the volume of the solid.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 3

To find the volume of the solid whose base is the shaded region ( R ) and whose cross sections, cut by planes perpendicular to the x-axis, are squares, you can follow these steps:

  1. First, determine the limits of integration along the x-axis. Since the shaded region is enclosed by the y-axis and the graphs of ( y = \sin(x) ) and ( y = \cos(x) ), find the intersection points of these two functions by setting them equal to each other: ( \sin(x) = \cos(x) ).
  2. Solve for ( x ) to find the intersection points. These will give you the limits of integration for ( x ).
  3. Once you have the limits of integration, integrate the difference between the functions ( \sin(x) ) and ( \cos(x) ) squared from the lower limit to the upper limit. This will give you the area of each square cross section.
  4. Finally, integrate the area of each cross section with respect to ( x ) from the lower limit to the upper limit to find the volume of the solid.

Using calculus, the volume ( V ) of the solid can be expressed as:

[ V = \int_{a}^{b} [(\sin(x))^2 - (\cos(x))^2] , dx ]

Where ( a ) and ( b ) are the limits of integration along the x-axis, determined from the intersection points of ( y = \sin(x) ) and ( y = \cos(x) ).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7