5.6b Notes - Total Area Recall If f β‰₯ 0 on a b the area between f and the x – axis is b Question What does ∠f x dx represent when f dips

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Total Area Recall: If f β‰₯ 0 on [ a, b ] , the area between f and the x – axis is ________________________. b Question: What does ∫ f ( x ) dx represent when f dips below the x – axis? a b ∫ f ( x ) dx = _________________________________________ a = __________________________‐_____________________________ What if we want to find the total area between the graph of f and the x – 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 ∫ f ( x ) dx, ∫ f ( x ) dx, ..., ∫ f ( x ) dx xn x1 3. Then total area = ∫ x2 f ( x ) dx + a ∫ b f ( x ) dx + ... + x1 ∫ f ( x ) dx xn Example 1: Find the total area between y = βˆ’ x 2 βˆ’ 2 x and the x – axis on the interval [‐3, 2]. Total Change The integral of a rate of change is the total change from a to b. ∫ b a F β€² ( x ) dx = F ( b ) βˆ’ F ( a ) Recall from physics: s(t ) = displacement v(t ) = sβ€²(t ) = velocity a(t ) = vβ€²(t ) = sβ€²β€²(t ) = acceleration Then ∫ 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 βˆ’ 2t βˆ’ 8 1 ≀ t ≀ 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, Ο€ ] . 4 Step 1: Sketch a graph of the functions. b Step 2: Set up the integral: ∫ ( 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 βˆ’ x 2 and y = βˆ’ 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 – axis and y = x βˆ’ 2 . Example 6: Find the area enclosed by y = x βˆ’ 1 and y 2 = 2 x + 6 . ...
View Full Document

This note was uploaded on 10/02/2011 for the course MATH 2214 at Virginia Tech.

Ask a homework question - tutors are online