Volume of Solid Bounded by Circle and Plane

In summary, the homework statement is to find the volume of the solid based on the interior of the circle, capped by the plane z=x. The attempt at a solution is to use the double integral over the circular region and integrate the function f(x,y)=x which in polar coordinates would be rcos(theta).
  • #1
tix24
25
0

Homework Statement



find the volume of teh solid based on the interior of the circle, r=cos(theta), and capped by the plane z=x.

Homework Equations

The Attempt at a Solution



i have drawn out the circle of equation r=cos(theta). I think that since z=x and is above the region, we have to use the double integral over this circular region. and integrate the function f(x,y)=x which in polar coordinate for would be rcos(theta).

so my train of thought is the following:

(∫ dtheta )(∫ (rcos(theta))rdr
where the limits of integration are for ∫ dtheta 0 to 2π
for ∫ (rcos(theta)rdr are from 0 to rcos(theta)

im not sure if my integrals are set up correctly any help regarding this problem would be very much appreciated. I have been stuck on this for a while
 
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  • #2
It looks good except for your limits on the ##\theta## integral.
 
  • #3
So how would I go about finding the limits for the r integral? I also think that the function which I'm integrating is incorrect.
 
  • #4
I didn't notice the mistake in the limits on the ##r## integral as well.

Suppose you wanted to calculate the area of the circle. What integral would you set up for that?
 
  • #5
vela said:
I didn't notice the mistake in the limits on the ##r## integral as well.

Suppose you wanted to calculate the area of the circle. What integral would you set up for that?
to calculate area by double integral i would have to integrate over the constant 1; my bounds of integration would be from r=0 to r=cos(theta) and i would have the second integral from theta=-Pi/2 to theta=Pi/2 for the theta integral
 
  • #6
Good. Now to get the volume, instead of integrating the function 1, you want to integrate the height of the solid as a function of ##r## and ##\theta##.
 
  • #7
So since the circle is capped be the plane, we have to integrate over the function f (x,y)=x which in polar coordinates is rcos (theta). I'm still not a 100% sure if this is correct or not.
 
  • #8
tix24 said:
So since the circle is capped be the plane, we have to integrate over the function f (x,y)=x which in polar coordinates is rcos (theta). I'm still not a 100% sure if this is correct or not.
You're doing fine. What integral do you get?
 
  • #9
So I set it up the following way:
The integral for theta went from -pi/2 to pi/2 and the integral for r went from 0 to cos (theta) the function which I integrated was rcos (theta) the result was pi/8

Can anybody confirm this?
 
  • #10
tix24 said:
So I set it up the following way:
The integral for theta went from -pi/2 to pi/2 and the integral for r went from 0 to cos (theta) the function which I integrated was rcos (theta) the result was pi/8

Can anybody confirm this?
Doesn't sound right, though I've not checked in detail. I meant, what integral expression do you get?
 
  • #11
my
haruspex said:
Doesn't sound right, though I've not checked in detail. I meant, what integral expression do you get?

my integral expression was as follows: integral dtheta (bounds from -pi/2 to pi/2) integral rdr (bounds from r=0 and r=cos(theta) and the expression which i integrate is x. which i wrote as x=rcos(theta)
 
  • #12
tix24 said:
mymy integral expression was as follows: integral dtheta (bounds from -pi/2 to pi/2) integral rdr (bounds from r=0 and r=cos(theta) and the expression which i integrate is x. which i wrote as x=rcos(theta)
Ok, that's correct, and I do get pi/8. I hadn't expected the 1/3 to get cancelled.
 

What is the definition of volume?

Volume is the amount of space occupied by an object or substance.

How do you calculate the volume of a solid?

The volume of a solid can be calculated by multiplying the area of the base by the height of the solid.

What are the units used to measure volume?

The units used to measure volume depend on the system of measurement being used. In the metric system, the most common units are cubic meters (m³) and liters (L), while in the imperial system, cubic feet (ft³) and gallons (gal) are commonly used.

Can the volume of an irregular shaped solid be calculated?

Yes, the volume of an irregular shaped solid can be calculated using the displacement method. This involves immersing the solid in a known volume of liquid and measuring the change in volume.

Why is finding the volume of a solid important?

Finding the volume of a solid is important in many scientific and engineering applications, such as designing structures, calculating density, and determining the amount of a substance needed for a reaction. It is also a fundamental concept in mathematics and helps us understand spatial relationships.

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