Question

What is the approximate volume of a slice perpendicular to the x-axis? Consider the region bounded by
x2 + y2 = 64 with x ≥ 0, and y ≥ 0
. A solid is created so that the given region is its base and cross-sections perpendicular to the x-axis are squares. Set up a Riemann sum

Answer 1

I get (8-x)^2(Delta x) but its wrong

Answer 2

@mathmale can you help me please?

Answer 3

I'd suggest that this problem would be easier to visualize and to solve if you'd at least make a rough sketch up front. First, you might draw a circle of radius 8, and then draw in a sample "slice" or two perpendicular to the x-axis.

Answer 4

Looking at that slice, contained within the circle, what's a formula for the length of that slice? alternatively, would you care to defend your own formula (8-x)?

Answer 5

okay i was solving for y and graphing that not actually graphing a circle like you said

Answer 6

I overlooked the fact that we are restricted to the first quadrant.
A sketch is not mandatory, but would be useful in visualizing what's happening in this problem situation.

Answer 7

Your proposed method, to solve the equation of the circle for y, will work. Unfortunately, the correct result is not 8-x. Care to try again? Solve x^2 + y^2 = 64 for y^2 and then for y.

Answer 8

x^2+y^2=64
y^2=64-x^2
y=sqrt(64-x^2)
y=8-x ?

Answer 9

the length of each slice is dependent upon y, and y itself is a function of x.

y=sqrt(64-x^2) is fine. But you cannot just take the sqrt of 64 and subtract the sqrt of x^2.

Try again.

Answer 10

so y= sqrt(64-x^2)

Answer 11

Actually, there's nothing more to do; you already have the correct result, which is
y=sqrt(64-x^2). Leave it as is; you cannot reduce or simplify it further.

So, what is the area of the individual slice? the thickness?

the volume?

Answer 12

area would be (sqrt(64-x^2))^2 which equals 64-x^2

Answer 13

volume is (64-x^2)(Delta x)

Answer 14

That's good. Why don't we check your expression for y by letting x=8? What would y be in that case?
What if x were 0? What would y be in that case?

Answer 15

y = 0
and y = 64

Answer 16

Check that 64, please. What about that sqrt operator in your formula for y?

Answer 17

y =8

Answer 18

Hint: your (correct) formula for y was y=Sqrt(64-x^2).

Answer 19

meaning the integral is from 0 to 8 right?

Answer 20

Excuse me. Right. But I was asking you for the lengths of the slices. the shortest is 0; the longest is 8.

Now, have you done Riemann Sum problems before?

Answer 21

yes but only worked on them briefly

Answer 22

Wish we had time to start with an easier problem.

Let me ask you, at this point: What is the integrand in this case (that is, what is the quantity to be integrated)?