1. 7.2 Areas Between Curves

2. Area Region R is bounded by the curves y = 2 – x 2 and y = -x. Sketch region R. R What is the area of region R?

3. Process To find the area between curves: 1. Sketch the region defined in the problem. 2. Connect the curves with either a vertical strip (dx) or a horizontal strip (dy). A strip that always connects the two curves will allow you to find the area without breaking up integrals. 3. Write an expression for the length of the rectangular strips. Vertical Strips: Length = Top curve – Bottom curve Horizontal Strips: Length = Right curve – Left curve NOTE: IF YOU USE A dy STRIP, YOU MUST SOLVE THE CURVE FOR x IN TERMS OF y. 4. Add rectangular strips together by setting up an integral using your expression. 5. Find points of intersection. NOTE: If using a dx, use the x-coordinates of intersection. If using a dy, use the y-coordinates of intersection.

4. Area What is the area of region R? dx Using a dx strip because it always connects the two curves. Length of dx strip = top – bottom y = 2 – x 2 y = -x = (2 – x 2 ) – (-x) = –x 2 + x + 2 Intersection Bounds??? 2 – x 2 = –x –x 2 + x + 2 = 0 –1(x + 1)(x – 2) = 0 –1(x 2 – x – 2) = 0 x = –1x = 2

6. Example Find the area bounded by y = e x, y = e 2, and the y-axis. Strip? dx Length? Bounds? x = 0 and intersection (e 2 = e x  x = 2)

7. Example Find the area between the two curves x = y 2 – 4y and y = x bounded by the x-axis. Strip? dy Length?Right – Left Bounds? y = 0 and intersection (y 2 – 4y = y  y = 5)

8. Homework Section 7.2 (#1-25 odd, 27-42 multiples of 3, 48)