# 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?

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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.

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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:

- 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) ).
- Solve for ( x ) to find the intersection points. These will give you the limits of integration for ( x ).
- 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.
- 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) ).

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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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