Unformatted text preview: Total Area Recall: If f â¥ 0 on [ a, b ] , the area between f and the x â axis is ________________________. b Question: What does â« f ( x ) dx represent when f dips below the x â axis? a b â« f ( x ) dx = _________________________________________ a = __________________________â_____________________________ What if we want to find the total area between the graph of f and the x â axis on the interval [ a, b ] ? A1= A2= A3= Then the total area= To do this with an actual function f: 1. Find all zeroes of f on [ a, b ] by setting f ( x ) = 0 and solving for x. 2. If zeroes occur at x1 < x2 < ... < xn , evaluate the following integrals: x1 x2 a x1 b â« f ( x ) dx, â« f ( x ) dx, ..., â« f ( x ) dx xn x1 3. Then total area = â« x2 f ( x ) dx + a â« b f ( x ) dx + ... + x1 â« f ( x ) dx xn Example 1: Find the total area between y = â x 2 â 2 x and the x â axis on the interval [â3, 2]. Total Change The integral of a rate of change is the total change from a to b. â« b a F â² ( x ) dx = F ( b ) â F ( a ) Recall from physics: s(t ) = displacement v(t ) = sâ²(t ) = velocity
a(t ) = vâ²(t ) = sâ²â²(t ) = acceleration Then â« v (t ) dt = b a Distance = Displacement = Example 2: Find the displacement and the distance traveled by a particle whose 2
velocity is measured by v ( t ) = t â 2t â 8 1 â¤ t â¤ 6 Area Between Two Curves Area between f and g on [a, b] = Example 3: Find the area between y = sec 2 x and y = sin x on [ 0, Ï ] . 4 Step 1: Sketch a graph of the functions. b Step 2: Set up the integral: â« ( top function-bottom function ) dx a Step 3: Evaluate. Regions enclosed by two curves: Example 4: Find the area of the region enclosed by y = 2 â x 2 and y = â x . Step 1: Find the points of intersection (set both equations = and solve for x) Step 2: Sketch the graph. Step 3: Set up and evaluate the integral as before. Integration with respect to y: Example 5: Find the area of the region in the first quadrant bounded above by y = x and below by the x â axis and y = x â 2 . Example 6: Find the area enclosed by y = x â 1 and y 2 = 2 x + 6 . ...
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