Areas between Curves

Vertical slices

Horizontal slices

Curves that cross

Areas with geometric meaning

The region between f(x) = 2 - x^2 and g(x) = x on [-2, 1] is sliced vertically. What integrand gives the strip height?

To find the area between two curves, the limits of integration are usually determined by what?

A region is sliced horizontally, bounded on the right by x = h(y) and on the left by x = k(y). What is the strip length?

Two curves cross at x = c inside [a, b], exchanging top and bottom positions. What must the area integral do?

The region between y = x and y = x^2 was computed as 1/6 by vertical slices and 1/6 by horizontal slices. What does the agreement demonstrate?

For the region between y = 2 - x^2 and y = x at the test point x = 0, which curve lies on top?

The area between y = x and y = x^2 from 0 to 1 equals the integral of (x - x^2) dx. What fraction is it? Answer like 1/6.

The area between y = 4 and y = x^2 on [-2, 2] has strip height 4 - x^2, always nonnegative there. At x = 0, what value does the integrand take? Answer with a number.

The curves y = 2 - x^2 and y = x intersect where x^2 + x - 2 = 0. How many intersection points enclose the finite region? Answer with a number.

When slicing horizontally, the integration variable is y and the differential is dy. As a two-letter string, what is that differential? Answer in lowercase.