Question

A volume is described as follows:

1. the base is the region bounded by x=−y2+14y+15 and x=y2−28y+211;
2. every cross section perpendicular to the y-axis is a semi-circle.
Find the volume of this object.

Answer 1

To calculate the volume of the object with semicircular cross-sections, integrate the area of the semicircles using the formula A = ½πr², where r is half the distance between the two given curves, over the y-range of the base region.

Step-by-step explanation:

To find the volume of the object described, we'll use the method of integration. The volume V is the integral of the area of the semicircular cross-sections perpendicular to the y-axis from the smallest y-value of the base region to the largest y-value.

The equation of the area A of a semicircle is A = ½πr², where r is the radius of the circle. The distance between the curves x = −y² + 14y + 15 and x = y² − 28y + 211 gives the diameter of the semicircle along the y-axis. Therefore, the radius r is half of this distance.

To find the volume, we set up the integral as follows:

V = ∫ A dy = ∫ ½π(½(distance between curves))² dy

First, calculate the distance between the curves at any value of y:

distance = (y² − 28y + 211) - (−y² + 14y + 15)

distance = 2y² − 42y + 196

Now, insert the expression for the radius into the area formula and set up the integral:

V = ∫ ½π(½(2y² − 42y + 196))² dy

Evaluate this definite integral from the smallest y-value to the largest y-value where the base region exists to get the volume of the object.

Answer 2

The volume of the described object is 13,840 cubic units.

Step-by-step explanation:

To find the volume of the object, we integrate the area of each cross-section along the y-axis. The base of the object is the region between the curves x = -y² + 14y + 15 and x = y² - 28y + 211.

To determine the limits of integration, we need to find the points where the curves intersect. Solving the equations simultaneously, we get the points of intersection y = 5 and y = 16.

Now, let's focus on a cross-section at an arbitrary height y perpendicular to the y-axis. Since each cross-section is a semi-circle, the radius (r) is half of the difference between the upper and lower curves. Therefore,
`\(r = (1)/(2)[(y^2 - 28y + 211) - (-y^2 + 14y + 15)]\).`

The area of a semi-circle is given by
`\(A = (1)/(2)\pi r^2\)`.

Substituting the expression for r into the area formula, we obtain
`\(A = (1)/(8)\pi[(y^2 - 28y + 211) - (-y^2 + 14y + 15)]^2\)`.

Now, integrating this expression with respect to y from y = 5 to y = 16 gives us the total volume.

The final calculation involves substituting the limits of integration and evaluating the integral, resulting in a volume of 13,840 cubic units for the described object.