Question

The base of the solid is the triangle enclosed by x + y = 11, the x-axis, and the y-axis. The cross sections perpendicular to the y-axis are semicircles. Compute the volume of the solid. (Use symbolic notation and fractions where needed.) V=

Answer 1

The final integral for the volume is V = ∫_{0}^{11}(π*((11 - y)/2)^2)/2 dy, which yields the volume when evaluated. To find the volume of the semicircular cross-sectioned solid, calculate the radius of each cross-section as a function of y, use the area formula for a semicircle, and integrate this area along the y-axis from 0 to 11.

Step-by-step explanation:

To compute the volume of the solid whose base is a triangle enclosed by x + y = 11, the x-axis, and the y-axis, with semicircular cross sections perpendicular to the y-axis, we use the equation for volume V = A∙h, where A is the area of the cross-section and h is the height.

The height, in this case, is the distance along the y-axis, which varies from 0 to 11 because the base triangle spans these values on the y-axis.

Step-by-step calculation:

1. First, we solve for x in terms of y using the equation x + y = 11 which gives us x = 11 - y.
2. The length of the base of the semicircle is equal to x, which varies with y. So, the radius r of the semicircular cross-section at height y is (11 - y)/2.
3. To find the area A of a semicircle, we use the formula A = (π∙r²)/2. Plugging in the radius, we get A(y) = (π*((11 - y)/2)^2)/2.
4. The volume of the solid is the integral of A(y) with respect to y from 0 to 11. Compute this integral to find the volume V.

The final integral for the volume is V = ∫_{0}^{11}(π*((11 - y)/2)^2)/2 dy, which yields the volume when evaluated.

Answer 2

The volume of the solid can be computed by integrating the areas of the cross sections perpendicular to the y-axis. The cross sections are semicircles, and their radii can be determined from the distance between the line x + y = 11 and the y-axis. The integral of the areas of the cross sections yields the volume of the solid.

Step-by-step explanation:

The volume of the solid can be computed by integrating the areas of the cross sections perpendicular to the y-axis. Since the cross sections are semicircles, the area of each cross section can be calculated using the formula for the area of a semicircle: A = (π/2)r^2, where r is the radius of the semicircle. To find the radius of each cross section, we need to find the distance between the line x + y = 11 and the y-axis. Since the base of the solid is the triangle enclosed by x + y = 11, the x-axis, and the y-axis, we can find the vertices of the triangle by setting x and y equal to zero in the equation x + y = 11. This gives us two points: (0, 11) and (11, 0). The distance between these points is the base of the triangle, which is equal to 11 units. Therefore, the radius of each cross section is half of the base, which is 11/2. Now we can compute the volume of the solid by integrating the areas of the cross sections from 0 to 11, and multiplying by the height of the solid. Since the height is not given in the question, we cannot calculate the exact volume. However, we can set up the integral as follows:

V = ∫((π/2)(11/2)^2) dy from 0 to 11.