Please show all work. 1. Find the area of the region bounded by the curves y=sin

Question

Please show all work.

1. Find the area of the region bounded by the curves

y=sinx, y=cosx, x=0, x= pi/4

2. Find the volume of the solid of revolution when the region from problem 1 is revolved about the x axis.

3. Find the volume of the solid of revolution when the region from problem 1 is revolved about the y-axis.

4. Find the coordinates of the centroid of the region from problem 1.

5. Find the arc length of the curve

y=x^(3/2), 0< or = x < or = 1.

6. Find the area of the surface of revolution when the region from problem 1 is revolved about the x-axis.

7. Find the area of the surface of revolution when the curve given by parametric equations x=5cost, y=sint 0 < or = t < or = pi/2, is revolved about the x-axis and about the y-axis.

Explanation / Answer

1)

You need to find the intersection points.

This is when sinx = cosx, or tanx = 1

x = pi/4 or 5pi/4

Then the area between the curves is

INT (sin(x) - cos(x)) dx from x = pi/4 to x = 5pi/4

= [-cos(x) - sin(x)] from x = pi/4 to x = 5pi/4

= [-cos(5pi/4) - sin(5pi/4)] - [-cos(pi/4) - sin(pi/4)] =

2sqrt(2) units of area.

2)

If the axis of revolution is not a boundary of the plane region and the cross sections are taken perpendicular to the axis of revolution, you use the washer method to find the volume of the solid. Think of the washer as a

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