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 âˆ« f ( x ) dx represent when f dips below the x â€“ axis? a b âˆ« 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 âˆ« f ( x ) dx, âˆ« f ( x ) dx, ..., âˆ« f ( x ) dx xn x1 3. Then total area = âˆ« x2 f ( x ) dx + a âˆ« b f ( x ) dx + ... + x1 âˆ« f ( x ) dx xn Example 1: Find the total area between y = âˆ’ x 2 âˆ’ 2 x and the x â€“ axis on the interval [â€3, 2]. Total Change The integral of a rate of change is the total change from a to b. âˆ« b a F â€² ( x ) dx = F ( b ) âˆ’ F ( a ) Recall from physics: s(t ) = displacement v(t ) = sâ€²(t ) = velocity
a(t ) = vâ€²(t ) = sâ€²â€²(t ) = acceleration Then âˆ« 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 âˆ’ 2t âˆ’ 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: âˆ« ( top functionbottom function ) dx a Step 3: Evaluate. Regions enclosed by two curves: Example 4: Find the area of the region enclosed by y = 2 âˆ’ x 2 and y = âˆ’ 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 âˆ’ 2 . Example 6: Find the area enclosed by y = x âˆ’ 1 and y 2 = 2 x + 6 . ...
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 EDeSturler
 Velocity, Tier One, Scaled Composites, Scaled Composites White Knight, 2004 in spaceflight, Suborbital spaceflight

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