6.6 Area Between Two Curves 6.5 Evaluating Definite Integrals 6.6 Area Between Two Curves

Properties of the Definite Integral

Find the area of the region R under the graph of from x=－1 to x =1. Evaluating Definite Integrals I Example Find the area of the region R under the graph of from x=－1 to x =1.

AREAS BETWEEN CURVES Consider the region S that lies between two curves y = f(x) and y = g(x) and between the vertical lines x = a and x = b. Here, f and g are continuous functions and f(x) ≥ g(x) for all x in [a, b].

AREAS BETWEEN CURVES As we did for areas under curves in Section 6.3, we divide S into n strips of equal width and approximate the i th strip by a rectangle with base ∆x and height .

AREAS BETWEEN CURVES Thus, we define the area A of the region S as the limiting value of the sum of the areas of these approximating rectangles. The limit here is the definite integral of f - g.

REVIEW The Definite Integral

Thus, we have the following theorem for area: AREAS BETWEEN CURVES Thus, we have the following theorem for area: The area A of the region bounded by the curves y = f(x), y = g(x), and the lines x = a, x = b, where f and g are continuous and for all x in [a, b], is:

AREAS BETWEEN CURVES Example Find the area of the region bounded above by y = bounded below by y = x, and bounded on the sides by x = 0 and x = 1.

Step 2 Formulate the definite integral Step 3 Find the value AREAS BETWEEN CURVES Solution: Step 1 Sketch the figure Step 2 Formulate the definite integral Step 3 Find the value

AREAS BETWEEN CURVES Example Find the area of the region enclosed by the parabolas y = x2 and y = 2x - x2.

AREAS BETWEEN CURVES From the figure, we see that the top and bottom boundaries are: yT = 2x – x2 and yB = x2

AREAS BETWEEN CURVES

properties

AREAS BETWEEN CURVES The area between the curves y = f(x) and y = g(x)（continuous）and between x = a and x = b is:

3 Vertical (horizontal) lines 4 Formula Summary 1 Draw the image 2 Judgment 3 Vertical (horizontal) lines 4 Formula

AREAS BETWEEN CURVES Exercise Find the area enclosed by the line y = x -1 and the parabola y2 = 2x + 6.

AREAS BETWEEN CURVES It would have meant splitting the region in two and computing the areas labeled A1 and A2. The method used in the example is much easier.

Some regions are best treated by regarding x as a function of y. AREAS BETWEEN CURVES Some regions are best treated by regarding x as a function of y. If a region is bounded by curves with equations x = f(y), x = g(y), y = c, and y = d, where f and g are continuous and f(y) ≥ g(y) for c ≤ y ≤ d, then its area is:

AREAS BETWEEN CURVES

AREAS BETWEEN CURVES Thus,

Find the area enclosed by 1.y=|x| and y=x2 - 2 2. y=x2 and x= y2 AREAS BETWEEN CURVES Exercises Find the area enclosed by 1.y=|x| and y=x2 - 2 2. y=x2 and x= y2