7.1 Area of a Region Between Two Curves

1. 7.1 Area of a Region Between Two Curves

2. Example Find the area between these two curves from x = -1 to x = 2. Consider a very thin vertical strip. The length of the strip is: or Since the width of the strip is a very small change in x, we could call it dx.

3. Since the strip is a long thin rectangle, the area of the strip is: If we add all the strips, we get:

5. Area The formula for the area between curves is: vertical representative rectangle (vertical strip)

6. Example Find the area between the curves and between x = -2 and x = 0.

7. Example Find the area between these two curves from x =0 to x = 4. If we try vertical strips, we have to integrate in two parts: We can find the same area using a horizontal strip. Since the width of the strip is dy, we find the length of the strip by solving for x in terms of y.

8. length of strip width of strip

9. Sometimes, it’s easier to express x as a function of y. • Must know which function on the right!! • Every thing must be written in terms of y Horizontal representative rectangle (horizontal strip)

10. 1 General Strategy for Area Between Curves Sketch the curves. Decide on vertical or horizontal strips. (Pick whichever is easier to write formulas for the length of the strip, and/or whichever will let you integrate fewer times.) 2 3 Write an expression for the area of the strip. (If the width is dx, the length must be in terms of x. If the width is dy, the length must be in terms of y. 4 Find the limits of integration. (If using dx, the limits are x values; if using dy, the limits are y values.) 5 Integrate to find area.

11. Examples Find the area between the curves : 1) 2) 3) 4)