Double Integrals over General Regions

Slices replace the grid

Reading bounds off a region

When one slicing needs help

Regions in applications

For the region between $y = x^2$ and $y = x$ with $0 \le x \le 1$, which integral computes $\iint_D x\, dA$?

Evaluating $\int_0^1 \int_{x^2}^{x} x\, dy\, dx$ gives which value?

A vertical slice through a region meets it in two separate segments for some values of $x$. What is the standard response?

For the region between $y = x^2$ and $y = x$, the inner pass $\int_{x^2}^{x} 1\, dy$ computes the slice length. Give the result as a polynomial like x-x^3.

A plate occupies the region between $y = x^2$ and $y = x$ for $0 \le x \le 1$ with constant density $2$. Give its mass, as a fraction like 1/3.