Volume of revolved triangle

Question

Use calculus to find the volume of the following solid S: The base of S is the triangular region with vertices (0,0), (3,0), and (0,2). Cross-sections perpendicular to the y-axis are semicircles.

When I draw the solid I see a half-cone. Then I find the volume using the disc method and I get 3π. But the autograder says it's wrong. Using the geometry formula for the volume of a half-cone, I still get 3π. So I'm thinking maybe I'm not drawing the picture correctly.

Answer 1

Step 1: Graph the base of the solid.

In order to determine the volume of this solid, we will need to start by plotting the base, using the given vertices: (0,2), (0,0) and (3,0), This triangular region represents the base of our solid. Note that, if we picture taking a cross section of this solid by cutting horizontally (perpendicular to the y-axis), then the diameter of each semicircle that we cut will have a diameter that extends from the y-axis all the way until it runs into the hypotenuse of the triangular base. This tells us that the length of the diameter will depend upon (or be defined by) the equation of the line that forms the hypotenuse of the triangular base.

Step 2: Determine the equation of the line that forms the hypotenuse of the triangular base using two of the points we were given: (0,2) and (3,0):

Slope (m) = -2/3 and y-intercept (b) = 2 this gives us the linear equation:

y = (-2/3)x + 2

* this equation will define (represent) the length of the diameter of our semicircle cross sections

Step 3: Write the Area function for the cross sections in terms of y (this is because each cross section will be cut perpendicular to the y-axis!).

This is done in two parts. First, we need the area formula for a semicircle:

A = (1/2)*pi*r²

Second, we need to substitute an expression for r (radius) into our area formula that is in terms of y. We know that the diameter can be represented with the equation we wrote in Step 2 (y = (-2/3)x + 2), so now we just need to get that equation in terms of y (just rearrange to isolate y). This gives us:

x = (-3/2)(y - 2) where x = the Length of the Diameter of semicircle cross sections

Altogether, this gives us our Area Function in terms of y:

A = (1/2)*pi*[(1/2)(-3/2)(y - 2)]² *notice that I multiplied the diameter expression by 1/2 to get the radius

Since we know we'll be taking the integral of the area function to get the volume of our solid, we can simplify the area function a bit to make our work easier:

A = (pi/2) (9/16) (y² - 4y + 4)

A = (9*pi/32) (y² - 4y + 4)

Step 4: Find the Volume of the solid. We'll use the formula: Volume of Solid = Integral (from 0 to 2- we need to use y- values since we're perpendicular to y-axis) of A(y) dy.

V = (9*pi/32) * Integral (from 0-2) of (y² - 4y + 4) dy

= (9*pi/32) [(y³/3) - 2y² + 4y] evaluating from 0 - 2, we get:

= (9*pi/32) [(8/3) - 8 + 8] -8 and 8 cancel out, giving us:

= (9*pi/32) (8/3)

= (3/4)*pi OR (3*pi)/4

Hope that helps!

Answer 2

Step 1: Graph the base of the solid.

In order to determine the volume of this solid, we will need to start by plotting the base, using the given vertices: (0,2), (0,0) and (3,0), This triangular region represents the base of our solid. Note that, if we picture taking a cross section of this solid by cutting horizontally (perpendicular to the y-axis), then the diameter of each semicircle that we cut will have a diameter that extends from the y-axis all the way until it runs into the hypotenuse of the triangular base. This tells us that the length of the diameter will depend upon (or be defined by) the equation of the line that forms the hypotenuse of the triangular base.

Step 2: Determine the equation of the line that forms the hypotenuse of the triangular base using two of the points we were given: (0,2) and (3,0):

Slope (m) = -2/3 and y-intercept (b) = 2 this gives us the linear equation:

y = (-2/3)x + 2

* this equation will define (represent) the length of the diameter of our semicircle cross sections

Step 3: Write the Area function for the cross sections in terms of y (this is because each cross section will be cut perpendicular to the y-axis!).

This is done in two parts. First, we need the area formula for a semicircle:

A = (1/2)*pi*r²

Second, we need to substitute an expression for r (radius) into our area formula that is in terms of y. We know that the diameter can be represented with the equation we wrote in Step 2 (y = (-2/3)x + 2), so now we just need to get that equation in terms of y (just rearrange to isolate y). This gives us:

x = (-3/2)(y - 2) where x = the Length of the Diameter of semicircle cross sections

Altogether, this gives us our Area Function in terms of y:

A = (1/2)*pi*[(1/2)(-3/2)(y - 2)]² *notice that I multiplied the diameter expression by 1/2 to get the radius

Since we know we'll be taking the integral of the area function to get the volume of our solid, we can simplify the area function a bit to make our work easier:

A = (pi/2) (9/16) (y² - 4y + 4)

A = (9*pi/32) (y² - 4y + 4)

Step 4: Find the Volume of the solid. We'll use the formula: Volume of Solid = Integral (from 0 to 2- we need to use y- values since we're perpendicular to y-axis) of A(y) dy.

V = (9*pi/32) * Integral (from 0-2) of (y² - 4y + 4) dy

= (9*pi/32) [(y³/3) - 2y² + 4y] evaluating from 0 - 2, we get:

= (9*pi/32) [(8/3) - 8 + 8] -8 and 8 cancel out, giving us:

= (9*pi/32) (8/3)

= (3/4)*pi OR (3*pi)/4

Hope that helps!

Answer 3

the function to integrate is pi times the square of y = (-3/2)x + 2 from x=0 to x=3

pi* integral ; [(-3/2)x + 2]^2 , x=[0,3]

pi * integral [ (9/4)x^2 - 6x + 4] , x=[0,3]

pi * [ (3/4)x^3 - 3x^2 + 4x], x=[0,3]

pi * [ 81/4 - 27 + 12]

pi * [ 81/4 - 15]

pi * [81/4 - 60/4]

pi * [21/4]