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#38682 - Math - Calculus

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.

Solution:

$$\Omega = \{(x, y) \mid (0 \leq x \leq 1) \text{ and } (0 \leq y \leq x^2)\}$$

Mass:

$$\begin{array}{l} M = \iint_{\Omega} xy \, dx \, dy = \int_{0}^{1} \left(\int_{0}^{x^2} xy \, dy\right) dx = \int_{0}^{1} x \left(\int_{0}^{x^2} y \, dy\right) dx = \int_{0}^{1} x \cdot \left(\frac{y^2}{2}\right) \Bigg|_{0}^{x^2} dx = \frac{1}{2} \int_{0}^{1} x^5 dx \\ = \frac{1}{12} (x^6) \big|_{0}^{1} = \frac{1}{12}. \end{array}$$

Answer: $M = \frac{1}{12}$.