Question

A plane sheet of material is bound by the curve y = x^2 from x = 0 to x = 1 the x-axis
and the line x = 1. If the mass per unit area (density) of the sheet is xy find the mass of
the sheet

Answer 1

Answer on Question #40754 – Math - Vector Calculus

A plane sheet of material is bound by the curve $y = x^2$ from $x = 0$ to $x = 1$ the x-axis and the line $x = 1$. If the mass per unit area (density) of the sheet is xy find the mass of the sheet.

Solution:

It is known that the mass of the plane sheet can be evaluated with the formula

$$m = \iint_{D} \rho(x, y) \, dx \, dy$$

where $\rho$ is the density of the sheet

$$\text{So, } m = \iint_{D} xy \, dx \, dy = \int_{0}^{1} x \, dx \int_{0}^{x^2} y \, dy = \int_{0}^{1} x \frac{y^2}{2} \int_{0}^{x^2} dx = \int_{0}^{1} \frac{x^2}{2} dx = \left. \frac{x^6}{12} \right|_{0} = \frac{1}{12}$$

Answer:

The mass of this plane sheet is $\frac{1}{12}$.