Question

How do you do this question?

Answer 1

V = 2000r³/3

Explanation:

We know that the base is a circular disk, so it creates a circle on the xy plane. It would be in the form x² + y² = r². In other words x² + y² = (5r)². Let's isolate y in this equation now:

x² + y² = (5r)²,

x² + y² = 25r²,

y² = 25r² - x²,

y = √25r² - x² ---- (1)

Now remember that parallel cross sections perpendicular to the base are squares. Therefore Area = length^2. The length will then be = 2√25r² - x² --- (2). Now we can evaluate the integral from -5r to 5r, of [ 2√25r² - x² ]² dx.

`\int _(-5r)^(5r)\:\left[\:2√(\left(25r^2\:-\:x^2\right))\:\right]\:^2\:dx\\=\int _(-5r)^(5r)4\left(25r^2-x^2\right)dx\\\\= 4\cdot \int _(-5r)^(5r)25r^2-x^2dx\\\\= 4\left(\int _(-5r)^(5r)25r^2dx-\int _(-5r)^(5r)x^2dx\right)\\\\= 4\left(250r^3-(250r^3)/(3)\right)\\\\= 4\cdot (500r^3)/(3)\\\\= (2000r^3)/(3)`

As you can see, your exact solution would be, V = 2000r³/3. Hope that helps!