I need help setting up the equation.

The portion of the graph y = ex
between x = 0 and x = ln 3 is rotated around the y axis to form a container. The container is filled with water. Use n = 4 subintervals and midpoints to approximate the work required to pump all of the water out over the side of the container.(Distance is measured in meters, the density of water is 1000 kg/m3, and use 9.8 m/s2 for g.)

I know you suppose to use the method of disk to find the volume.But then I need to know the work done...

So x=ln(y)

Would it be something like
V=pi*(ln(y))^2*delta y ?

Please help, thanks

18898.05075 is the correct answer accepted by Webassign.

whoops the graph is y=e^x

Well, well, well, looks like you've got a little math pickle on your hands. Don't worry, I've got just the right amount of silliness to help you out!

Now, let me just put on my math clown hat and get into the zone. *Honk honk!*

To find the work required to pump the water out, we first need to find the volume of the container, right? And you're absolutely correct, we can use the method of disks to do that.

The volume of each disk is given by the formula V = πr^2 * h, where r is the radius and h is the height. But hold on, we need to express r and h in terms of y, not x.

Since x = ln(y), we can rewrite it as y = e^x. But remember, we're dealing with the range between x = 0 and x = ln 3. So when x = 0, y = e^0 = 1, and when x = ln 3, y = e^(ln3) = 3.

Now, let's find the radius and height in terms of y. The radius, r, is simply the x-value, which in this case is ln(y). The height, h, is the change in x, which is equal to Δx.

Using your nifty equation for the volume, V = π(ln(y))^2 * Δy, we're on the right track! But, but, but, don't forget to express Δy in terms of y. Since you have n = 4 subintervals, Δy = (3 - 1) / 4 = 0.5.

Once you've found the volume of each disk using the equation above, it's time to find the total work. The work done is equal to the force applied (which is the weight of the water) multiplied by the distance.

The weight of the water is given by the formula W = m * g, where m is the mass and g is the acceleration due to gravity. And since density (ρ) equals mass divided by volume, we can rewrite the equation as W = ρ * V * g.

Now, we know the density of water is 1000 kg/m^3, and g is 9.8 m/s^2.

So, the total work done is W = (1000 kg/m^3) * (V) * (9.8 m/s^2).

And there you have it, my friend! You’ve got your equation for the total work. Now, all that's left to do is plug in those numbers and crunch away!

Remember, math can be a bit slippery at times, but just keep juggling those equations until you reach the answer you're looking for. You've got this! Good luck! *Throws confetti*

To find the work required to pump all of the water out over the side of the container, we first need to find the volume of water in the container. As you mentioned, we can use the method of disks to find the volume.

The equation y = ex represents a curve on the graph. Since we are rotating this portion of the curve around the y-axis, the resulting shape will be a solid disk. The radius of each disk will vary based on the value of y.

To find the radius of each disk, we need to express y in terms of x. You correctly identified this as x = ln(y). Solving for y, we get y = e^x.

Now, let's calculate the volume of each disk using the formula you provided, V = π * (ln(y))^2 * Δy. However, we need to modify it slightly to match the form of the equation we derived earlier. So, let's substitute y = e^x into the equation:

V = π * (ln(e^x))^2 * Δy
= π * x^2 * Δy

We can further simplify the expression because x is the same as ln 3 when x = ln 3. So, x^2 will be equal to (ln 3)^2.

Now that we have the volume of each disk, we can sum them up to find the total volume of water in the container. Since we are using n = 4 subintervals, we can split the interval from x = 0 to x = ln 3 into four equal subintervals.

Let's say Δx is the width of each subinterval. In this case, Δx = (ln 3 - 0) / 4.

We can calculate the midpoints of the subintervals by taking the average of the left and right endpoints of each subinterval. For example, the midpoint of the first subinterval is (0 + Δx/2), the midpoint of the second subinterval is (Δx + Δx/2), and so on.

Now, let's calculate the volume of water in each corresponding subinterval. Multiply the area (π * (ln 3)^2) by the width (Δx) for each subinterval.

Once you have the volume of water in each subinterval, sum them up to get the total volume of water.

Finally, to calculate the work required to pump all of the water out over the side of the container, you need to multiply the total volume by the weight of the water. The weight of water is equal to the volume multiplied by the density of water (1000 kg/m3) and the acceleration due to gravity (9.8 m/s2).

So the work done will be: Work = Volume * density * g

Plug in the values you have and perform the necessary calculations to get the final answer.

The portion of the graph

y = ex
between x = 0 and x = ln 3 is rotated around the y axis to form a container. The container is filled with water. Use n = 4 subintervals and midpoints to approximate the work required to pump all of the water out over the side of the container. Give your answer in decimal form.