# Finding the Area Between Curves

How do you find the area of a region bounded by two curves? I'll consider two cases.

Suppose the region is bounded above and below by the two curves("top" and "bottom"), and on the sides by and .

Think of dividing the region up into vertical rectangles. The height of the typical rectangle is , while the thickness is . The area of a typical rectangle is

To find the total area, integrate to add up the areas of the little rectangles:

The in the integral is a reminder that I want "top" and "bottom" expressed in terms of x.

Suppose the region is bounded on the sides by two curves ("left" and "right"), and on the top and bottom by and .

Think of dividing the region up into horizontal rectangles. The height of the typical rectangle is , while the thickness is . The area of a typical rectangle is

To find the total area, integrate to add up the areas of the little rectangles:

The in the integral is a reminder that I want "right" and "left" expressed in terms of y.

Example. Find the area of the region bounded by and the x-axis.

Example. Find the area of the region bounded by and .

Example. Find the area of the region bounded by and .

Example. Find the area of the region bounded by and .

The curves intersect at the (approximate) values -1.10710, 0.83757, and 0.26959.

Example. Find the area of the region between the curves and from to .

Example. Find the area of the region between the curves and from to .

The area of the region under is

The area of the semicircle is .

Hence, the area of the region between the curves is

Example. Find the area of the region bounded by and .

The region consists of two pieces. For the left-hand piece, the top curve is and the bottom curve is . For the right-hand piece, the top curve is and the bottom curve is .

I need to find where the curves intersect. Solve the equations simultaneously:

The intersections are at , , and .

The area is

Example. Find the area of the region bounded above by the curves and and below by the x-axis:

(a) Using vertical rectangles.

(b) Using horizontal rectangles.

The curves intersect at , .

Using vertical rectangles, I need two integrals:

Using horizontal rectangles, I only need one integral: