8-2 Areas in the plane.

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Presentation transcript:

8-2 Areas in the plane

Area Between Curves To find the area between a curve and the x-axis, we can simply integrate. Now we will learn about area between two curves. The idea is still the same. – Partition & sum as rectangle widths go to 0. Def: Area Between Curves If f and g are continuous with f (x)  g (x) through [a, b], the area between curves f (x) and g (x) from a to b is the integral of [ f – g ] from a to b. **Surprise! It is very helpful to sketch the graphs first!**

Ex 1) Find the area of the region between y = sec2 x and y = sin x from x = 0 to y = sec2 x y = sin x

Area enclosed by Intersecting Curves *We need to find the intersection points.* Ex 2) Find the area of the region enclosed by the parabola y = 2 – x2 and the line y = – x. Intersect at: 2 – x2 = –x 0 = x2 – x – 2 0 = (x – 2)(x + 1) x = 2, –1 y = 2 – x2 y = – x

Using a Calculator Ex 3) Find the area of the region enclosed by the graphs of y = 2cos x and y = x2 – 1 Enter eqtns under Y1 and Y2 2nd CALC  5: intersect Enter, enter, enter Quit 2nd ANS  STO under A Repeat for 2nd intersection point and store under B y = 2cos x y = x2 – 1 Math 9: fnInt  4.995 Y1 – Y2

Partitioning a Region Ex 4) Find the area of the region R in the first quadrant that is bounded above by and below by the x-axis and the line y = x – 2. I + II II I

Higher x-values (on top) Sometimes it’s easier to integrate with respect to y instead of with respect to x. Ex 5) Find the area of the region in Ex 4 by integrating with respect to y. Higher x-values (on top)

Using Geometry Ex 7) Find the area of the region in Ex 4 by subtracting the area of the triangular region from the area under the square root curve. Whole area under curve – triangle

Ex 6) Find the area of the region enclosed by the graphs of y = x3 and x = y2 – 2. (Use any strategy you wish – wrt x or wrt y) If by x, have to partition area at x = a. x = a If by y, don’t have to partition Find intersection points for upper and lower bounds  4.21

homework Pg. 390 # 3, 17, 21, 34, 35 Pg. 399 # 3–45 (mult of 3)