Section 7.2a Area between curves.

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Presentation transcript:

Section 7.2a Area between curves

Area Between Curves Suppose we want to know the area of a region that is bounded above by one curve, y = f(x), and below by another, y = g(x): Upper curve First, we partition the region into vertical strips of equal width and approximate each strip as a rectangle with area a b Note: This expression will be non-negative even if the region lies below the x-axis. Lower curve

Area Between Curves Suppose we want to know the area of a region that is bounded above by one curve, y = f(x), and below by another, y = g(x): Upper curve We can approximate the area of the region with the Riemann sum a b The limit of these sums as is Lower curve

Definition: Area Between Curves If f and g are continuous with throughout [a, b], then the area between the curves y = f(x) and y = g(x) from a to b is the integral of [f – g] from a to b,

Guided Practice Find the area of the region between and from to . Now, use our new formula to find the enclosed area: First, graph the two curves over the given interval:

Guided Practice Find the area of the region between and from to . First, graph the two curves over the given interval:

Guided Practice Find the area of the region enclosed by the parabola and the line . The graph: To find our limits of integration (a and b), we need to solve the system Algebraically, or by calculator: a b

Guided Practice Find the area of the region enclosed by the parabola and the line . Because the parabola lies above the line, we have The graph: a b units squared

Guided Practice Find the area of the region enclosed by the graphs of and . The graph: To find our limits of integration (a and b), we need to solve the system Solve graphically: a b Store the negative value as A and the positive value as B.

Guided Practice Find the area of the region enclosed by the graphs of and . Note: The trigonometric function lies above the parabola… The graph: Let’s evaluate this one numerically… Area: a b units squared

Guided Practice 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 . The graph of R: Area of region A: 2 (4,2) B 1 A 1 2 3 4

Guided Practice The graph of R: Area of region B: 2 (4,2) B 1 A 1 2 3

Guided Practice The graph of R: Area of R = Area of A + Area of B: 2 (4,2) B 1 A Units squared 1 2 3 4

Guided Practice Find the area of the region enclosed by the given curves. First, graph in [–2,12] by [0,3.5] Intersection points: Break into three subregions:

Guided Practice Find the area of the region enclosed by the given curves. First, graph in [–2,12] by [0,3.5] Intersection points: Break into three subregions: