I would like to make sure my answer is correct:
Question: the base of a solid is the triangular region with the vertices (0,0), (2,0), and (0,4). Cross sections perpendicular to the x-axis are semicircles. Find the volume of this solid.
My Work: ∫[0,2] (pi/2)(r^2)dx
r = (-2x+4)/2 = (-x+2)
r^2 = x^2-4x+4
(pi/2) x^3/3 - 4x^2/2 +4x with limits regarding -1 and 1.
My Answer: 4pi/3
I would greatly appreciate if you could let me know if my answer is correct just so I know I am practicing this properly! Thank You!!
UPDATE** I meant with limits regarding 0 and 2!!
Looks good to me.
To verify if your answer is correct, let's go through the steps of calculating the volume of the solid.
First, we need to find the equation of the radius of the semicircles as a function of x. The base of the solid is the triangular region with vertices (0,0), (2,0), and (0,4). The height of the triangle is y = 4, so the equation for the line connecting the vertices (0,0) and (0,4) is x = 0. The equation for the line connecting the vertices (0,0) and (2,0) is y = 0. So the equation for the line that connects both is x - 2y = 0, which can be rewritten as x = 2y/2 or x = y.
Since the cross-sections perpendicular to the x-axis are semicircles, the radius of each semicircle is given by the equation of the line we just found, x = y.
To calculate the volume, we integrate the area of each cross-section over the interval [0,2] with respect to x:
∫[0,2] (π/2)(r^2) dx
Substituting r = x, we have:
∫[0,2] (π/2)(x^2) dx
Evaluating the integral, we have:
(π/6) x^3 |[0,2]
Evaluating at the limits, we get:
(π/6)(2^3) - (π/6)(0^3)
(8π/6) - (0)
(4π/3)
Therefore, the correct answer is 4π/3.
Your answer of 4π/3 is indeed correct! Well done!