# The region under the curves #y=cosxsqrtsinx, 0<=x<=pi/2# is rotated about the x axis. How do you sketch the region and find the volumes of the two solids of revolution?

A sketch is provided:

graph{cos(x)sqrt(sin(x)) [-1.5, 1.5, -1, 4]}

So, the volume of the entire solid will be given by the integral of this function within the bounds, or:

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

To sketch the region under the curve (y = \cos(x)\sqrt{\sin(x)}) for (0 \leq x \leq \frac{\pi}{2}), you first need to analyze the behavior of (\cos(x)) and (\sqrt{\sin(x)}) separately in this interval.

- For (\cos(x)), as (x) varies from (0) to (\frac{\pi}{2}), (\cos(x)) starts at (1) and decreases to (0).
- For (\sqrt{\sin(x)}), as (x) varies from (0) to (\frac{\pi}{2}), (\sqrt{\sin(x)}) starts at (0) and increases to (1).

So, the region under the curve will be bounded by the x-axis and the curve itself.

To find the volumes of the two solids of revolution, you'll need to divide the region into two parts.

- For the first part, rotate the region under the curve about the x-axis. This forms a solid with a "washer" shape, as there's a hole in the middle.
- For the second part, rotate the region under the curve about the x-axis up to the curve itself. This forms a solid with a cylindrical shape.

To find the volume of each solid, you'll need to integrate using the disk (washer) or cylindrical shell method, depending on the part. For the first part, the washer method involves subtracting the volume of the inner circle from the volume of the outer circle. For the second part, you'll use cylindrical shells.

Once you set up the integrals, integrate them with respect to (x) from (0) to (\frac{\pi}{2}), and evaluate to find the volumes of the two solids of revolution.

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

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.

- How do you use the midpoint rule to estimate area?
- How do you find the volume of the parallelepiped with adjacent edges pq, pr, and ps where p(3,0,1), q(-1,2,5), r(5,1,-1) and s(0,4,2)?
- What is the area enclosed between the two polar curves: #r = 4 - 2cos 3theta# and #r = 5# ?
- How do you compute the volume of the solid formed by revolving the given the region bounded by #y=sqrtx, y=2, x=0# revolved about (a) the y-axis; (b) x=4?
- How do you find the area between the curves #y=x^2-4x+3# and #y= 3+4x-x^2#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7