Integrating a volume

• February 22nd 2010, 07:33 PM
Latszer
Integrating a volume
We just started today and a problem I have is...

Write a Riemann sum and then a definite integral representing the volume of the region, using the slice shown in the figure below. Evaluate the integral exactly. Use your work to answer the questions below.

What is the approximate volume of the slice with respect to y?

So I got l/14=(7-y)/7 and then I get 20y deltay is the volume of the slice but it is a circle, where would I put pi into this?

Thanks for the help,

Tyler
• February 22nd 2010, 10:04 PM
DBA
The area of the cut is a rectangle A=2x*10
$
V = \int_{a}^{b}A(y)dy
$

$
V = \int_{a}^{b} 2x * 10 dy = 20 \int_{a}^{b} x dy
$

So, you need to write the x in terms of y, since you have dy

The point (x,y) is on the circel
$x^{2} + y^{2} = r^{2}$
$x^{2} + y^{2} = 7^{2}$

So,
$x = \sqrt {49-y^{2}}$

$
V = 20 \int_{0}^{7} \sqrt {49-y^{2}} dy
$