Question

Find the mass of the lamina with the given density. Enter an exact answer, do not use decimal approximation. Lamina bounded by x = y2 and x = 49, p(x, y) = 4y2 + 8x + 46 = The mass is

Answer 1

To find the mass of the lamina, integrate the density function over the region between the curves x = y^2 and x = 49. Set up the double integral, perform the integration, and obtain the exact mass of the lamina.

Step-by-step explanation:

To find the mass of the lamina, we need to calculate its volume and multiply it by its density. The lamina is bounded by the curves x = y^2 and x = 49. We can integrate the density function p(x, y) = 4y^2 + 8x + 46 over the region between the two curves to find the total mass.

First, we find the limits for the integration:

For the lower limit, y = 0 and x = y^2 = 0. So, the lower limit is x = 0.

For the upper limit, y = sqrt(x) and x = 49. So, the upper limit is x = 49.

Now, we can set up the integral:

m = ∫∫[R] p(x, y) dA

where R is the region between the curves, dA is the infinitesimal area element, and m is the mass. Integrating with respect to x first, we have:

m = ∫[0 to 49] ∫[0 to sqrt(x)] (4y^2 + 8x + 46) dy dx

After performing the integration, we get the exact mass of the lamina.