The area of a region under a curve is greater than the area of an inscribed rectangle and less than the area of a circumscribed rectangle. Intuitively, we understand that somewhere “between” the inscribed and circumscribed rectangles there is a rectangle whose area is precisely equal to the area of the region under the curve:

The area of a region under a curve is greater than the area of an inscribed rectangle and less than the area of a circumscribed rectangle. Intuitively, we understand that somewhere “between” the inscribed and circumscribed rectangles there is a rectangle whose area is precisely equal to the area of the region under the curve:

The area of a region under a curve is greater than the area of an inscribed rectangle and less than the area of a circumscribed rectangle. Intuitively, we understand that somewhere “between” the inscribed and circumscribed rectangles there is a rectangle whose area is precisely equal to the area of the region under the curve: