1 6.1 Areas in the Plane Mrs. Kessler

2 How can we find the area between these two curves? We could split the area into several sections, A,B,C, and D, use addition and subtraction to figure it out, but there is an easier way. A B D C

3 Consider a very thin vertical strip or rectangle. 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.

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

5 How many strips are there? How can we add all the strips?

6

7 The formula for the area between curves is:

8 Another example f(x)=cos(x) g(x)=0.5x-1 A = -  /2 b =  /2 =  +2 

9 Page 418 #20 f(x) = - x 2 +4x+2 g(x) = x+2 First find points of intersection, x= a and x = b Use TI 83+ x = 0, 3 = 4.5

10 FOR CHECKING ONLY Page 418 #20 Using TI FOR CHECKING ONLY Y1 f(x) = - x 2 +4x+2 Y2 g(x) = x+2 First find points of intersection Use TI 83+, 2 nd Calc Intersect x = 0, 3 Why is the answer negative?

11 What if graphs intersect each other? f(x) = - x 2 +4x+2 g(x) = x+2 On the interval [0,4] 1. First determine the point(s) of intersection. 2. Determine where f(x) > g(x). 3. Set up multiple integrals.

12 Intersecting graphs: f(x) = - x 2 +4x+2 g(x) = x+2 On the interval [0,4] =6.3333

13 General Strategy for Area Between Curves: 1 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.) For today vertical only. Sketch the curves. 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. 

14 Horizontal area between two curves

15 What is the best way to determine the area between these two functions?

16 If we try vertical strips, we have to integrate in two parts.

17 There is another way We can find the same area using a horizontal strip and only have to integrate once. Since the width of the strip is dy, we find the length of the strip by solving for x in terms of y.

18 How do we adjust this formula? What are the limits of integration?

19 length of strip width of strip

20 Example 2. Use horizontal strips to find the area between these two curves. HOW? First, sketch.

21 Example 2. Use horizontal strips to find the area between these two curves. 1.Sketch 2.Solve for x, if not already done so. 3.Find the points of intersection. 4.Determine the width of the horizontal strips. 5.Determine the limits of integration. 6.Set up the integral. 7.Solve.

22 Example 2. Use horizontal strips to find the area between these two curves.

23 Example 2. Use horizontal strips to find the area between these two curves. = 4.5

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25 General Strategy for Area Between Curves: 1 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.) Sketch the curves. 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. 