7.2 Area Between Two Curves

I. Area Between Curves A. ) Graphical Representation

B.) Assuming f(x) and g(x) are continuous on [a, b] and f(x) ≥ g(x)… Approximate area of the region:

Now, take the limit as our greatest change in x approaches zero and…

C.) Procedure: 1.) Construct a figure, writing the equations of the boundary curves. 2.) Determine the pts. of intersection. 3.) Decide whether to use vert. or hor. rectangles as the typical element of area. DRAW and LABEL this REPRESENTATIVE RECTANGLE.

3. ) (cont. ) We need 1 consistent expression for the length 3.) (cont.) We need 1 consistent expression for the length. (Avoid two points on the same curve.) 4.) Set up integral(s) and evaluate. 5.) Note: If you get the set-up and cannot integrate, change variable or use calculator if allowed.

II. Examples A.) Ex. 1 - Find the area bounded by the intersection of the curves Set-up: Points of Intersection -

Solution-

B.) Now, let’s find the same area by partitioning the y-axis, or by using horizontal rectangles.

B.) Ex. 3 - Find the area bounded by the intersection of the curves Set-up: Points of Intersection -

Solution-

C.) Ex. 4 - Find the area bounded by the intersection of the curves Set-up:

Solution (in terms of x)-

Set-up:

Solution (in terms of y)-